Approximations for the Study of Drift Boundaries in the Magnetosphere

VOL. 80, NO. 25 JOURNAL OF GEOPHYSICAL RESEARCH SEPTEMBER 1, 1975

Approximationsfor the Study of Drift Boundariesin the Magnetosphere

M. G. KIVELSON

Institute of Geophysicsand Planetary Physics, University of California Los Angeles, Calijbrnia 90024

D. J. SOUTHWOOD

PhysicsDepartment, Imperial College,London, England

An approximaterelation betweenenergy and equatorialradial distancefor a particleof pitch angleae, moving adiabatically in a dipole magneticfield is used to describethe Alfv6n layer for particlesof arbitrary pitch angle. The convectionelectric field is assumedto be uniform in the equatorialplane of the magnetosphere,and magnetic field lines are taken to be equipotentialsin the region traversed by the magnetosphericparticles being studied. Approximate solutionsin the high- and low-energylimits are given in a form directly applicable to the interpretation of measurementsat fixed particle energy and pitch angle. These resultsare particularly useful for electrons,since the approximations provide lower bounds to the exact solutions.The results are used to interpret recent Explorer 45 electron measurements of Williams et al.

INTRODUCTION terpret the substorm-associatedelectron fluxes reported The establishment or enhancement of an electric field recentlyby Williamset al. [1974] from Explorer 45 measurements. oriented from dawn to dusk across the magnetosphereis widely recognizedas a significantfeature of geomagnetically DEFINING EQUATIONS FOR THE ALFVI•N LAYER active periods [Axford, 1969; Mozer, 1971; Mcllwain, 1972; Heppner, 1973; Gurnettand Frank, 1973; Gurnettand Akasofu, An Alfvfin layer is definedfor a particle of magneticmoment 1974], and the resulti.nginward convectionand adiabatic ac- u as a flow line on which there is a stagnationpoint where celerationof tail plasma [Axford and Hines, 1961;Brice, 1967; electric field drifts and magneticdrifts exactlybalance. These DeForest and Mcllwain, 1971] contribute to the substorm- boundaries can be very complex for low-energy protons, associatedparticle flux enhancementsobserved in the quasi- whosemagnetic and corotation drifts can produceresonances, dipolar regions of the magnetosphere[Arnoldy and Chan, allowing them to penetrate into the inner magnetosphere 1969; Lezniak and Winckler, 1970; Konradi et al., 1973; Wil- [Chen, 1970]. Figure 1, adapted from Chen [1970], showsthe liamset al., 1974]. However, it is not possiblefor tail plasmato possiblecomplexity of these proton orbits. We will merely be convectedinto all portions of the near magnetosphere remark on the existenceof multiple stagnationpoints for such (L •< 10) becausethere are 'forbidden zones' into which parti- protons and develop approximations to describe cases for cles cannot penetrate by electric convection [Alfvdn and which the Alfvfin layer is geometricallysimple (cf. Figure 3). Ftilthammar, 1963;Schield et al., 1969]. The boundaryof the Convective flow conservesthe total energy including the forbidden zone is referred to as the.'Alfv•n layer,' and its potential energy of the assumeduniform dawn-dusk electric geometry is a sensitivefunction of the adiabatic invariants of field and of the corotation electric field. In the earth's dipole the particleswhose motion it characterizes[Chen, 1970, 1974] magneticfield the corotation potential is [2BoRE:•L-• in terms and of the magnitude of the convectionelectric field. Solutions of Bo,the equatorial field at 1 RE, and [2 = 2•r/day. Magnetic have been obtained for a number of representativecases for field lines are assumedto be equipotentialsin the region 90ø pitch angle particles in a dipole magnetic field [Chen, traversed by the particles being studied. Thus for convective 1970, 1974], but the quantitative resultshave been published flow, only for the midnight meridian. In this paper we derive some W + qERELsin 0 - (q/Iql) CL-• = const (1) approximatesolutions which can be usedto calculateproper- ties of Alfv6n layers in the high- and low-energylimits and wherefor the earth, C = I ql[2BORE •' • 90 keV. which apply to particlesof arbitrary equatorialpitch angle.As The azimuthal angle is measured counterclockwisefrom we shall see, the high- and low-energylimits provide a very midnight.The vanishingflow velocityat the stagnationpoint good guide to the behavior of electronsof arbitrary energy. requires The resultsare formulated as answersto the question,What OW cross-magnetosphereelectric field will producean Alfv6n layer OL = --qER•esin qO,-- (q/Iql)CL, -•' (2) at (L, ½) for a particlewhose energy at (L, ½) is W? If the cross- magnetospherefield is known, the relation obtained can be in- The variation of W with L for a chargedparticle whose initial verted to find the upper cutoff energy for particlesconvected equatorialpitch angle is sin ae.o is well approximatedby the inward from the tail to the equatorial point (L, ½)or, alter- expression natively,to determinethe L value of the boundarybetween closed and open drift orbits for a particle of energy W at azimuthal angle ½. The approximationobtained is usedto in- Wo-- v = 2.1q- 0.9 sin C•eq 0 (3) as demonstratedby Southwoodand Kivelson[1975]. Cowley Copyright ¸ 1975 by the American GeophysicalUnion. and Ashour-Abdalla[1974] have shown that for Lo/L < 5 this 3528 KIVELSONAND SOUTHWOOD: DRIFT BOUNDARYAPPROXIMATIONS 3529

Equation (7) can be solvednumerically for electronsof any energy,in which casethe only approximation is the use of (3). In the next section,however, we develop solutionswhich are useful in the high- and low-energylimits.

HIGH- AND LoW-ENERGY LIMITS In the high-energylimit, C/WL << 1, the corotation terms on the right-hand side are small and may be neglectedto lowestorder. This conditionis alwayssatisfied for proton orbits correspondingto casesfor which L8 is the larger solution of (5) on the dawn meridian; for the condition that the root be real, (vWs)•' > 4eERsC can be combinedwith the inequalityL8 > v Ws/(eERs) to give

LW L, W8 -- v W,

--<4 •4 since the stagnation point is always the point at which the Fig. 1. (Adapted from Chen [1970].) Complex orbits of protonsof Alfv6n layer is farthestfrom the earth. The zeroth-orderap- small magnetic moment t• with two stagnationpoints on the dusk meridian.The heavycurves pass through stagnation points at a and c proximation to y satisfies and bound regions in which closed orbits exist. The dashed lines cor- respondto open orbits. An exampleof a closedorbit is shownas a dot- v sin 4>yov+• + (v + 1)yff - 1 = 0 (8) ted curve. For v = 3, (8) reducesto the solution obtained in the high- energyestimate is accurateto betterthan 2% for aeq> 5ø. At energylimit by Alfv6n [Alfv•n and Fiiltharnrnar,1963]. Plots of aeq = 0 ø the energyestimate is lesssatisfactory, but the correct yoas a functionof sin 4>for v = 2 and v = 3 are givenin Figure value is obtainedby settingv -- 2. From (3) the stagnationcon- 2. Energeticprotons are describedby (8) with sin 4>-• -sin 4>. dition becomes The curves of Figure 2 indicate that when corotation can be neglected,the electron Alfv6n layers near dawn (or proton vW8 = qERsL8sin ½8+ (q/[ql)CL8-• (4) Alfv6n layers near dusk) are quite insensitiveto equatorial and cos48 = 0, as is evidentlyrequired by the symmetryof the system.The positionsof the stagnationpoints are v W8=1= [(v Ws)" -- 41qlER•Csin 4•,] L• = (5) 2qER• sin Thus electronsof energy W have one stagnationpoint on the dusk meridian,while protonsof energyW haveone stagnation point on the dusk meridian and two additional stagnation points on the dawn meridian if v W8 > (4qERsC) •/•'. (Conversely,for protons at fixed u there is one stagnation point on the dawn meridian and two additional ones on the dusk meridian if # < (C•'/4vqERs),surprisingly not an incon- sistentstatement.) The Alfv6n layerof interestfor high-energy protonsis the one which passesthrough the largerL8 at dawn, for this definesthe region within which closedorbits encircle the earth and outside of which orbits are open. It is im- mediatelyclear that for the usualmagnetospheric electric fields (E •.<3 kV •), protons of energy >(39/v) keV on the dawn meridian are always on closedorbits, for unlessL8 is lessthan --•10 Rs, the boundarylies outside the magnetopauseand if, in addition, E •< 3 kV Rs-•, (4) implies W8 > (39/v) keV. The locusof a boundarywhich passes through the stagnation point at L8 is found by settingthe total energyof (1)equal to its value at the stagnationpoint. Working nominally with Fig. 2. Approximatesolutions for the Alfv6n layer for electronsof electronswith q = -e and 48 = 3•r/2, we find pitch anglesae, = 90ø (v = 3) and ae, = 0 ø (v = 2). Diagrams(a) and (b) apply to electronsin the high-energylimit R/WL << I. The solu- W- W8 = eERs(L8 + L sin 4) + CL8-• - CL -• (6) tions for high-energyprotons are obtainedby setting½ = -½ and y = yo - (C/WL)by. Diagram (a) showsy0(•), the solutionof (8), while EliminatingeERs from (4) and (6), we obtainan equationfor y diagram(b) givesby(•) definedby (9). Diagrams(c) and l,d)apply to = L/L8 electronsin the low-energy limit, WL/C << I. Diagram (c)gives Y0'(•), the solutionof (1 la), which is independentof equatorialpitch vy"+• sin ½ + (v + l)ff' - 1 angle,and diagram(at) gives by'(•), definedby (1 lb). Under conditions discussedin the text, diagrams (c) and (at) can be adapted for low- = (C/LW)(I - 2y - y•' sin 4) (7) energy protons. 3530 KIVELSONAND SOUTHWOOD:DRIFT BOUNDARYAPPROXIMATIONS pitch angle but that both the asymmetryof the Alfv6n layers and their dependenceon equatorial pitch angle becomelarge in the direction of the stagnation point. On this side of the earth the particle energyis lowest,and convectionis naturally more important. One shouldnote that the fractionalchange of electron energy on a closedorbit betweenmidnight and dawn \ \ is of the order of 30% for 2 _< v _< 3. The correction of order C/WL to the solution of (7) is • \\ \\ / 1 -- 2yo • yo2sin4• by = (9) v(v -+- 1)yo•-•(yosin 4• -3- 1) wherey = Yo+ (C/WL)by. Valuesof by are plotted in Figure 2 • 2//////// x, / / and are seen to be small. The equivalent result for energetic \x protons is found by setting 4• = -4•, W = -W. To zeroth order in C/WL, the drift orbit of a particle on the Alfv6n layer is found by expressingthe stagnationcondition ((4) with C = 0)in terms of y as L = vW,yo(ck)/(IqlER•) (10) Fig. 3. Schematic representationof the relation between the boundary LA(½) (heavy curve) between closedand open orbits for a Hencea particleat longitude½ with energyW hasan orbit proton of energy W and the Alfv6n layersfor high-energyprotons given by (dashed curves). The diagram is in the equatorial plane with noon at the top of the page. Assumethat protons of magnetic moment tt• on the Alfv6n layer labeled 1 have energy W at point b. They loseenergy L= IqlWR as they drift acrossthe electricfield; henceat noon their energyis less than W. However, for some larger magnetic moment tt•, protons on if it is on an Alfv6n layer. the Alfvfin layer labeled2 will have energy W at noon (point a). LA(½) then is defined by points on Alfv•n layers of different tt. In the low-energylimit the corotationterm in (7) dominates, so an approximate solution of the form Yo' + L W/Cby' is given by first obtain an approximatesolution for E• and invert it to ob- sin OSy0'2 + 2y0' - 1 = 0 (11a) tain L•. In the high-energylimit we find from (4), (8), and (9) the result • Yot t•y'-- 2(yo'-- 1)[PYø'V+I sin4• q- (u q- 1)y0 'v- 1] W VYo C yo(1 -- Yo) eE.4R•:-- L (vq- 1 q-Vyo sin 40 q-•(yosin4•q- 1)

= , [vyo w g 2(1 q- Yo sin -- a T q-• (12) q-(v q- 1)yo'•-- 1] (11b) For a known cross-magnetospherefield, (12) can be solvedto The quantitiesYo' and by' are also plotted in Figure 2. The give L•. In these expressions,y0 is a function of ½ defined by result for low-energyprotons, found by setting W = -W, is (8) and plotted in Figure 2. The equivalent resultsfor high- useful only for the portion of the Allyfin layer near the dusk energyprotons are obtained by settingC = -C and ½ = -½ in stagnationpoint labeleda in Figure 1, becauseof the complex- (8) and (12). The parameters a and/5 for electronsof 0 ø and ities of the multiple-valued low-energy solutions. For 90 ø pitch angle versus sin ½ are shown in Figure 4 in units electrons,such complicationsdo not arise. which give eERe in keV. The dawn-dusk asymmetry of the boundary L• is much ELECTRIC FIELD PRODUCING AN ALFV!•N LAYER greater than the asymmetryof the Alfvfin layersthemselves, as FOR GIVEN (L, W, can be seen when corotation is ignored by comparing the As satelliteexperiments typically give information on parti- curvesfor y0(oS)in Figure 2 with the correspondingcurves for clesin a specificenergy range, it is of practicalvalue to be able a(OS)in Figure 4. The larger asymmetry of the boundary for to answerthe following questions.For a given value of W and fixed energy occurs becauseLA(½) crossesAlfv6n layers of in- aeq, what cross-magnetospherefield EA = EA(W, aeq;L, 4)) creasingly large # as it approaches the side of the would place the Alfv6n layer at (L, 05),or alternatively,for magnetospherewhere the kinetic energyof the particle species given values of W, a,q, and cross-tailpotential, what is the is smallest.These Alfvfin layers with large # are found at great locusLA = LA(W, a,q, E; 4)) of the boundarybetween closed distancesfrom the dipole center. and open orbits?Figure 3 showsschematically how the bound- The ratio of LA for 0 ø particles to LA Of 90ø particles is ary LA is related to the Alfv6n layers of high-energyprotons, significant,for it indicatesthat small pitch angle particlescan which conservemagnetic moment # in their drift. Within the penetratefurther than 90ø pitch angle particlesinto the dipole boundary LA an energeticproton observedwith energy W is field from a sourceregion in the tail. For example, near mid- on a closedtrajectory in the dipole field, whereasat positions night the boundary for energetic 0 ø particles lies about 25% beyond LA the trajectories of particles of the same energy further in than the 90ø boundary. The possibility of large must be open. energy dependent pitch angle anisotropies occurring in the As LA is defined by an equation of third or fourth order, region betweenthe boundariesLA(V = 3) and LA(V -- 2) has in- dependingon the pitch angle of interest,it is more sensitiveto terestingconsequences for the possibledevelopment of wave- small errors in the approximate solution than EA is. Hence we particle instabilities, a possibility which is being investigated KIVELSONAND SOUTHWOOD:DRIFT BOUNDARYAPPROXIMATIONS 3531

be convectedin to smallerradial distancesthan protons.(Swift

3 [1971] has commented on this feature of convective flow in a reversedgeomagnetic dipole field.) Correspondingly, electrons can gain more energythan protons during inward convection, and this may play a role in producing the intense energetic v=3 electron flux observed in the dipole field region on Jupiter [Opp, 1974, and referencescited therein]. 0 For low-energy electrons the values of EA and LA are obtained analogouslyby use of (11) in the limit WL/R << 1, which yields •' 5o eE.•RE=•5 Yo '2 LW (1 yo--[-•(yo yo'• • sin-- 1) v=2 g5 .' w 3) -I.O -0.5 0 0.5 1.0 --Z •+ L Sin• whereYo' is definedby (1 la). Curvesfor B' and a' are given in Fig. 4. Parametersfor the evaluationof the electricfield required Figure 5. Equation (13) can be solved for L to give LA for for an electron of energy W to lie on an Alfv6n layer at (L, ½) in the known E. The low-energyapproximation for protons, useful high-energylimit. The parametersa and/• of (12) areplotted for v = 3, near the dusk stagnation point, is obtained by setting W = aeq = 9.0ø, and v = 2, aeq = 0 ø. The solutionsfor protons,ap and •W. are obtained from av(½) = a(-4), •v(½) = -/•(-½), and eERE = The coefficientsa and B are remarkably closeto the coeffi- (apW/L) + (•p/L•'). cients a' and B', as is seenin Figure 6, where curvesof (a - a')/a and (B' - B)/B versussin ½ are plotted for the casee = 3; by M. Ashour-Abdallaand S. W. H. Cowley(personal com- the valuesfor e = 2 are of the sameorder. This suggeststhat munication, 1974). the linear expressionis a goodapproximation for any electron The corotation contributionsto EA are positive for electrons energy(Chen's [1970] resultsfor intermediateenergy protons becausethe addition of the corotation velocity effectivelyin- indicate that this could not be good for protons). We can in creasesthe magnetic angular velocity relative to the electric fact show that approximations(12) and (13)provide lower field drift velocity.In order to obtainbalance at the stagnation bounds to EA. From (4) we obtain point, the electricfield must increase.By similar reasoning,if the electric field is fixed, the stagnationpoint must move out- eER• vWs C •+1 W 2 C ward to decreasemagnetic drift effectsrelative to electric field -- L, + L•• = vy • + y Z• (14) drift effects.As Ls increases,the Alfv6n layer movesout at all With • = eER•L2/C and p = WL/C we have 4. The boundaryLA lies at larger radial distancesfor electrons, • = pvy•+l + y2 whose corotation and magnetic drifts supplementeach other, where y is an implicit function of p by virtue of (7). Now than for protons,whose corotation and magneticdrifts oppose each other. In the Jovian magnetosphere,where the dipole orientation is oppositeto that of the earth, the corotation drift • = vy + [pv(v+ 1)y• + 2y]Op opposesthe magnetic drift for electrons,which can therefore From (7),

0••, =k P [2 + pv(vy•sin•+ 1)y•-•](1+2y-- 1 and so

y(1 -- y•) • 1 q-y sin4• (15)

I I I I I I I I I I

75 I i i ! '1I ! i i I f ! i i 'l I i i i .08 •

.06

.04

-I.O -0.5 0 0.5 Sin • .0•

Fig. 5. Same as Figure 4 for the low-energylimit. The parameters 0 [ , ] • • I , , • [ I , , , • ! , ,. , • and •' of [13) are plottedfor v = 3, a,, = 90ø, and v = 2, a,, = 0 ø. -I.0 -0.5 0 0.5 1.0 (/•' is independentof the valueof v.) The solutionsfor protons,ap' and •v', are obtained from av'(½) = -a'(½), •v'(½) = •'(½), and eERE = S•nc• (•,'/L •-) + (,,' W/L). Fig. •. •a = (a - a')/•, and • ---(/•' - /•)//• for ,, = 3532 KIVELgON AND SOUTHWOOD: DRIFT BOUNDARY APPROXIMATIONS

clearly relevant to observationsof the type discussedby 0p•' • \•-•/ 1 + y sinqb Kivelson and Southwood[1975], who interpreted the asym- metric distribution of particle flux at fixed energyobserved in Since0 < .0 _< 1, it follows that 0r//0p > 0 and 0•'r//Op•' > 0 the early part of a geomagnetic storm [Frank, 1970]. For (there is no problem with the limits y -, 1, sin 0 -• -1). Thus analysisof this type of experimental data we have found the the electricfield satisfies (0E/O IF)[L.• > 0, whichimplies that approximate expressionsof the precedingsection to be a useful E(W:, L, qb)< E(W•, L, O) if W: < W• and that O:E/OW •' > alternative to exact numerical solutions.As an example of 0, which indicatesthat the curve of E as a function of W (at further experimentalobservations which can be interpreted fixedL and 0) is concaveup. Since(13) specifiesthe tangentto consistentlyby usingthe argumentsof this paper, we consider this curve at W = 0, it always lies below the curve; since the complex substorm-associatedincreases of electron flux (12) specifiesthe asymptote,it must also lie below the curve. reported by Williams et al. [1974]. The electron flux was Hence the electricfield EA(W, L, 4) satisfies measuredby the Explorer 45 satelliteas it moved through the aW premidnight magnetospherenear apogeeat L = 5.5. (In this + < eEA(W, L, ck) spatial region, Alfv6n layersand precipitationboundaries are likely to be approximately coincident.) Williams et al. note that althoughthe high-energy(35-560 keV) flux peaks arrived + • < eEA(W, L, ck) at the satellite with a dispersion consistentwith magnetic gradientdrift to the satellitefrom a sourcenear midnight,the The low-energy(high-energy) approximation lies closestto highest-energyparticles arriving earliest, the low-energy the exact solution when W < Wc(W > We), where (1.5-10.8 keV) fluxes arrived in the reverse order, the lowest- a'W• energy particles arriving first at the satellite. Williams et al. L + _ L +Z suggestthat the sourceof the low-energyelectrons could be the plasma sheet, which is convectedin by an electric field for or, in termsof Aa = (a -- a')/a and aB = (B' - B)/B (plotted which they obtain a crudeestimate of 0.7 kV R• -•. They reject in Figure 6), this interpretation,because they note that particletrajectories L Wo = calculated by Mcllwain [1972] for a model electric field developedto interpretparticle measurements at geostationary Similar arguments can be in9oked to show that orbit do not penetratedeeply enoughto bring plasmasheet O• O•E electronsto the Explorer 45 orbit. Noting, however,that the •- <0 >0 minimum penetrationdistance LA(½) is a rather sensitivefunc- OL w OL• tion of the cross-tailfield, we can determinehow large an From theseinequalities it can be proved that the valuesof L as electric field is required to allow electrons of W < 8 keV to a function of E, W, • obtained from (12) and (13) provide convect inward to the Explorer 45 orbit and then examine lower boundsto La(E, W, •) for electronsof all energies. other aspectsof the experimentalobservations to seewhether they are reasonablyconsistent with the assumedelectric field. INTERPRETATION OF EXPLORER 45 ELECTRON OBSERVATIONS Relevantfeatures of the Williamset al. [ 1974]measurements In the previoussection the contour La(•) has been defined in the low-energychannels are given in Table 1. If the signifi- as the boundary of the region inaccessibleto particleswhose cant aspectsof these observationsare to be interpreted in sourceis in the tail and whose local energyis W. Fluxes terms of convectionboundaries, we must postulatean electric measuredat fixed energy W may changerapidly with radial field in whichthe Explorer45 orbit intersectsthe cutoffbound- distanceacross this boundary,especially in the early stagesof ariesgiven by (13) for W < 5.3 keV and does not intersectthe a storm or substormbefore diffusion by nonadiabaticproc- cutoff boundaryfor W = 8.0 keV. The requirementis met for essesacts to smoothdiscontinuities. Similarly, temporalvaria- 1.8 kV R• -• < E < 2.2 kV R• -•, and such electric fields are in tionsof the flux at fixedenergy may occur quite independently the range usually associatedwith substorms[Mozer, 1971; on the two sidesof the boundary.The decouplingof the spatial Heppner, 1973; Gurnett and Frank, 1973]. In Figure 7 we and/or temporalvariations of the flux acrossthe boundary La(•) helps to identify the existenceof an Alfv6n boundary TABLE 1. Positionsof Initial IncreasesObserved in Low-Energy layer in analysisof measurements.Strong precipitationcan Electron Fluxes [Williams et al., 1974] and Predicted Intersections of the Explorer 45 Orbit also produceboundaries across which particle fluxeschange markedly, and Kennel [1969] has discussedthe coupling Observations Predictions for E = 1.9 betweenprecipitation and convectionboundaries. In identify- Energy ing a spatialboundary as an Alfv6n layer, the alternativepos- Range sibilitythat the boundaryis a precipitationboundary must be 8.0-10.8 ...... considered.Wolf[1970] has useddifferent modelsof the con- 5.3-7.1 5.3 -32 ø 5.5 -45 ø vectionelectric field to calculateelectron Alfv6n layers and 3.5-4.8 5.5 - 38 ø 5.4 - 51 o precipitationboundaries and has shownthat althoughthey 2.3-3.2 5.5 -40 ø 5.3 -55 ø may be well separatedon the day sideof the magnetosphere, 1.5-2.1 5.5 -47 ø 5.2 -58 ø Plasmapause 5,5 -48 ø 4.9 -63 ø they coincide closely over a large part of the nighttime out magnetosphere.He has also noted that the near coincidenceof Plasmapause 4.5 - 14ø 3.2 8 ø the boundaries is a consequenceof the fact that near the in stagnation point on the Alfv6n layer the plasma flows so The cutoff boundaries were obtained from (13) for the lowest- slowly that it has a high probabilityof being precipitated energy electrons in each channel in a field E = 1.9 kV RE-•. before it is convectedaway. Dots indicate that no rise was observed and that no rise was The applicationof an Alfv6nlayer analysis is probably most predicted. KIVELSONAND SOUTHWOOD: DRIFT BOUNDARY APPROXIMATIONS 3533

1.9kV/RE tion [Kivelsonand Southwood, 1975] with little az•imuthaldrift. Thusthese electrons would have reached the s,atellitemainly by beingconvected inward. For 1 Re of inwarddisplacement 900 from L -- 6.5 to L -- 5.5, adiabaticconvection leads to energy increasesof the order of 60%, and this couldeasily account for the peaksin the low-energychannels. The interpretationof the -75 ø arrival times of the low-energy particles has already been given. Turning to the proton measurements which are also reportedby Williamset al. [1974],we can merelynote that the -60 ø energeticprotons are found within their Alfv6n layersbut that

/ / I we have not analyzedthe lower-energyprotons because of the -450 -30 ø -150 0ø complexity of low-energy proton orbits previously noted. Fig. 7. Cutoff boundariesevaluated from (13) for E = 1.9 kV RE-• However, Smith and Hoffman [1974] have developedargu- for the lowest-energyelectrons in the low-energychannels of the Ex- ments with a philosophy similar to that of the arguments plorer 45 chargedparticle detector [Williams et al., 1974].The circled presentedhere in their interpretationof proton fluxesobserved numbers give the energy associatedwith each boundary, and the plasmapause(p.p.) is the calculatedboundary for W = 0. The dots by Explorer 45 during the beginning of several magnetic (from left to right) indicatepositions at whichflux beganto risein the storms in 1971 and 1972. energy channels 1.5-2.1, 2.3-3.2, 3.5-4.8, and 5.3-7.1 keV. The sub- CONCLUSIONS storm onset was at 2200 + I UT, and crossbars on the orbit are marked at every hour beginningat 1955 UT. The convection boundaries associated with the cross- mognetosphereelectric fields occurring during the early stage show the Explorer 45 orbit and the cutoff boundariesfor the of magneticstorms and substormsdepend strongly on particle lowest-energyparticles in eachenergy channel in a field of 1.9 energy and azimuthal position. The spatial and temporal kV Re -•. variationsof the particleflux may be largelydecoupled across In Table 1 we tabulate the intersections of the satellite orbit suchboundaries, so that sharpgradients or very differenttime with the calculated boundaries as well as with the calculated dependencemay be observedfor particleswhose energies lie plasmapause,identified on Explorer 45 by the saturation of an aboveor belowsome cutoff energy. We hopethat the simple electric field measurement[Maynard and Cauffman, 1973]. approximationsgiven in this paper will prove usefulin identi- Both the predictedboundary crossings and the observedflux fying such boundariesin experimentaldata. The simplicityof increasesoccur at increasinglongitude with decreasingenergy, theseresults isinevitably tied to thesimplicity of the'model we though there is a consistentdiscrepancy of the order of have assumed, and we feel that thei r main use should be as a 11ø-15ø betweenthe predictedand observedlongitudes. From rule of thumb guide,most powerful when applied to electrons. Figure 7 it can be determined that if the dawn-dusk electric Naturally, the further the magneticfield departsfrom a dipole fieldwere somewhat skewed [e.g., Volland, 1973], the crossings (and the further the electricfield departsfrom a uniform dawn- of the boundariesas well as the plasmapauseencounters would dusk field), the greater shouldbe the degreeof skepticismac- coincide more closely with the observed flux increases.The corded to our estimates. We would like to point out the highly idealizedmodel of this paper can at best apply only to usefulnessof consideringthe boundary LA(W, qb)as it was average behavior, so we have not attempted to adjust describedhere in future more sophisticatedwork. One should parameters to obtain better agreement with the observations. further bear in mind that the model fields are static and that, The Williams et al. [1974] observations can now be in- accordingly,there is an inherenttime lag in the completefor- terpreted by consideringhow electronsin the range 1.5-560 mation of an Alfv6n layer which is comparablewith the drift keV would respond to the expected combination of tail time• about the earth. magneticfield collapse[McPherron et al., 1973] and an ,--2 kV The analysisof the Williams et al. [1974] electronsin this Re -• dawn-dusk electric field associated with the substorm paper and recentstudies of proton fluxesduring the develop- with onset V, hour before the first flux increaseswere observed. ment phaseof magneticstorms [Kivelson and Southwood,1975; During substorms,significant increases ita the magneticfield Smith and Hoffman, 1974] show that drift boundariescan be are typicallyobserved within about 3 hoursof localmidnight significantand mustbe consideredin an,analysis of particle [Clauer and McPherron, 1974]. As Explorer 45 was initially behavior. During the later (main and recoverY)phases of outsidethis local time region and moved in local time at a rate s•tormsthe simplesharp boundary produced by a time in- of lessthan or approximatelyequal to the corotationangular dependentelectric field is no longer found, both becausenon- velocity,it did not encounterparticles which had beenlocally adiabaticprocesses become increasingly important over longer acceleratedby the changingmagnetic field. However, high- time intervalsand becauserepeated injection events implying energy electronsinitially within the region of magneticfield time-varyingelectric fields (a typeof nonadiabaticprocess) can increasesnear midnight and acceleratedat the substormonset moveparticle boundaries to newpositions. For thfsreason the encountered the satellite after drifting once around the energyand L dependentenhancements Ofenergetic proton flux earth, as suggestedby Williams et al. The assumedelectric field measuredduring the main phaseand recoveryphase of a would have some effecton the energeticelectrons but would storm by Konradiet al. [ 1973]cannot be interpretedin terms of not changethe picture in any qualitative way [Walker and an Alfv6n layer for a reasonableelectric field. Analogous Kivelson, 1975]. measurementsmade during the developmentphase of a storm For the low-energy electronsthe electric field which has should show some of the features which we have described and beenintroduced, and whichmay be thoughtto havedeveloped would be of great interest. 30 min to 1 hour before the substormonset [Mozer, 1971], Acknowledgments. The authors have benefitted from numerous would have simultaneouslyproduced convection and accelera- conversationswith S. W. H. Cowley,who discussed with them his own 3534 KIVELSON AND SOUTHWOOD: DRIFT BOUNDARYAPPROXIMATIONS

work on the Alfv6n layerand gaveadvice willingly. George Ioannidis Kennel, C. F., Consequencesof a magnetosphericplasma, Rev. and Daniel Kivelson kindly helped to clarify some points, and the Geophys.Space Phys., 7, 379, 1969. refereesmade valuable suggestions which improved the paper.One of Kivelson,M. G., andD. J. Southwood,Local time variations of parti- the authors(M.G.K.) wishesto thank the John SimonGuggenheim cle flux producedby an electrostatidfield in the magnetosphere,J. Memorial Foundation for its supportand J. W. Dungey for his en- Geophys.Res., 80, 56, 1975. couragementduring the tenure of her fellowship.This work is sup- Konradi,A., D. J. Williams,and T. A. Fritz, Energyspectra and pitch ported in part by NASA under contract NGL 05-007-004. Institute of angledistributions of storm time and substorm-injected protons, J. Geophysicspublication number 1399. Geophys.Res., 78, 4739, 1973. The Editor thanks A. J. Chen and R. A. Wolf for their assistance in Lezniak, T. W., and J. W. Winckler, Experimentalstudy of evaluating this paper. magnetosphericmotions and the accelerationof energeticelectrons during substorms,J. Geophys.Res., 75, 7075, 1970. Maynard, N. C., and D. P. Cauffman, Double-floatingprobe REFERENCES measurementson S•-A, J. Geophys.Res., 78, 4745, 1973. Mcllwain, C. E., Plasma convection in the vicinity of the Alfv6n, H., and C.-G. F•ilthammar,Cosmical Electrodynamics, Ox- geosynchronousorbit, in Earth's MagnetosphericProcesses, edited ford University Press, New York, 1963. by B. M. McCormac, D. Reidel, Dordrecht, Netherlands, 1972. Arnoldy, R. L., and K. W. Chan, Particle substormsobserved at the McPherron,R. L., C. T. Russell,and M.P. Aubry, Satellitestudies of geostationaryorbit, J. Geophys.Res., 74, 5019, 1969. substormson August15, 1968,9, Phenomenologicalmodel for sub- Axford, W. I., Magnetosphericconvection, Rev. Geophys.Space storms, J. Geophys.Res., 78, 3131, 1973. Phys., 7, 421, 1969. Mozer, F. S., Origin and effectsof electricfields during isolated Axford, W. I., and C. O. Hines, A unifyingtheory of high latitude magnetosphericsubstorms, J. Geophys.Res., 76, 7595, 1971. geophysicalphenomena and geomagnetic storms, Can. J. Phys.,39, Opp, A. G., Pioneer10 mission:Summary of scientificresults from the 1433, 1961. encounter with Jupiter, Science, 183, 302, 1974. Brice,N.M., Bulk motionof the magnetosphere,J. Geophys.Res., 72, Schield,M. A., J. W. Freeman, and A. J. Dessler,A sourcefor field- 5193, 1967. alignedcurrents at aurorallatitudes, J. Geophys.Res., 74, 247, 1969. Chen, A. J., Penetrationof low-energyprotons deep into the Smith, P. H., and R. A Hoffman, Direct observations in the dusk magnetosphere,J. Geophys.Res., 75, 2458, 1970. hoursof the characteristicsof the stormtime ring currentparticles Chen, A. J., Correction, J. Geophys.Res., 79, 5314, 1974. duringthe beginningof magneticstorms, J. Geophys.Res., 79, 966, Clauer, C. R., and R. L. McPherron,Variability of mid-latitude 1974. magneticparameters used to characterizemagnetospheric sub- Southwood,D. J., and M. G. Kivelson,An approximateanalytic storms, J. Geophys.Res., 79, 2898, 1974. descriptionof plasmabulk parametersand pitch angleanisotropy Cowley,S. W. H., and M. Ashour-Abdalla,Adiabatic plasma convec- underadiabatic flow in a dipolarmagnetospheric field, J. Geophys. tion in a dipole field: Variation of plasmabulk parameterswith L, Res., 80, 2069, 1975. Rep. 92131, Centre Nat. d'Etud. de Telecommun.,Issy-les- Swift, D. W., Possiblemechanisms for formationof the ring current Moulin•aux, France, July 1974. belt, J. Geophys.Res., 76, 2276, 1971. DeForest, S. E., and C. E. Mcllwain, Plasma clouds in the Volland, H., A semiempiricalmodel of large-scalemagnetospheric magnetosphere,J. Geo'phys.Res., 76, 3587, 1971. electricfields, J. Geophys.Res., 78, 171, 1973. Frank, L. A., Direct detection of asymmetricincreases of extrater- Walker, R. J., and M. G. Kivelson,Energization of electronsat syn- restrial 'ring current' proton intensitiesin the outer radiation zone, chronousorbit by substorm-associatedelectric fields, J. Geophys. J. Geophys.Res., 75, 1263, 1970. Res., 80, 2074, 1975. Gurnett, D. A., and S.-I. Akasofu, Electricand magneticfield obser- Williams, D. J., J. N. Barfield,and T. A. Fritz, Initial Explorer45 sub- vationsduring a substormon February24, 1970,J. Geophys.Res., storm observationsand electric field considerations,J. Geophys. 79, 3197, 1974. Res., 79, 554, 1974. Gurnett, D. A., and L. A. Frank, Observedrelationships between Wolf, R. A., Effectsof ionosphericconductivity on convectiveflow of electricfields and auroral particleprecipitation, J. Geophys.Res., plasma in the magnetosphere,J. Geophys.Res., 75, 4677, 1970. 78, 145, 1973. Heppner, J.P., High latitude electric fields and the modulations (Received August 28, 1974; related to interplanetarymagnetic field parameters,Radio Sci., 8, revised March 17, 1975; 933, 1973. acceptedApril 4, 1975.)