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 ar- bitrary 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 measure- ments. 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 or- bits 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 farthest from 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 stagna- tion 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