Dimensional Hybrid Code Simulations of a Relaxing Field Reversal

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. A9, PAGES 17,391-17,404, SEPTEMBER 1, 1994

Self consistent one-dimensional hybrid code simulations of a relaxing field reversal Alan Richardson • and SandraC. Chapman SpaceScienceCentre (MAPS), Universityof Sussex,Brighton,United Kingdom

Abstract.

We present the results of self-consistentone-dimensionalhybrid code

(kineticionsand fluid electrons)simulations of a dipolarizingfield reversal.Local ion energization processesare studied in terms of both the ion kinetics and bulk

plasmabehavior.We demonstrate evolutionof the systemthroughtwophases:(1) a quasi-static"trapping" phase,during whichinfiowingionsare trapped in the reversal

structureat early timesand (2) a dynamic"escape"phase,duringwhichtrapped

ionsescape fromthe reversal afteranincrease in/•1' Weperformsimulations with differentlinkingfieldcomponent (GSE B•) to reversingcomponent (GSE B•) ratios and show that this parameter controlsthe details of the ion energization. This

studysuggests a characteristic signaturein the magneticfield (a bipolarB•) and

bulkplasma parameters (anincrease in /•l) associated withthisbehavior which

should be observablein in situ data. Our results are applicable for times earlier than the midtail to ionosphereAlfv6n travel time.

mechanisms which occur are all local to the dipolariz-

1. Introduction

ing field line, enablingus to deducewhat energization can occur locally and by implication what must occur

At the onsetof a magneticsubstorm,magneticfield (e.g., at a reconnection site). linesin the near-Earth geotailundergorapid reconfigu- elsewhere We present the results of two simulations, one with a ration from a taillike geometryto a more dipolarstate weak linking field component and one with a stronger ("dipolarization").Characteristic featuresof dipolarlinking field. The values we use span the range of obization includerapid changesin the local field strength, theobservation of high-energy (• 1MeV) burstsof par- servedlinking field component. We demonstratethat ticles[e.g.,Honesel al., 1976],andthe arrivalof lower- the ratio of linking field to reversingfield components energyplasmaseenat around6RE (so called "injec- determinesthe details of ion energization,rather than tion events")[e.g.,Lennarlson el al., 1981].To date, the overall evolution of the relaxing reversal. We find that the reversingmagneticfield structureis only non-self-consistent studieshavebeenmade of ion capable of trappinginfiowingions. As the trappedions dynamicsduring dipolarization[e.g., Chapman,1993, oscillate acrossthe reversal,they gain net field-aligned 1994;Delcourt,1991;DelcourtandMoore,1992]. In this paper we presentone-dimensional hybrid code accelerationvia a mechanismapproximately modelled simulationsof a dipolarizingfield reversal.In order to as first-order Fermi acceleration. The parallel pressure study the consequences of dynamicreconfiguration of of the trapped population increasesdue to individual the magnetotail magneticfield at substormonset we ion acceleration, and once it has become sufficiently choose as our initial condition an out-of-equilibrium large to strongly perturb the bulk plasmaconfigurafield reversal. This work is significantin that unlike tion, the trapped population escapes. The escaping previousstudiesof ion energizationduringdipolariza- population may be in pressurebalanceor may have an dependingon the sizeof the linking field. tion [e.g.,Delcourt,1991;DelcourtandMoore,1992], overpressure, After the escape of the initially trapped population, a we can self-consistentlyexamine the evolution of local field structure and ion dynamics. The ion energization continualprocessevolvesin whichionsinitially located outside of the reversal convectin, are trapped, acceler-

ate, and escapeen masse.A characteristicbipolar By 1Now at TessellaSupport Services,Abingdon,United Kingdom.

Copyright 1994 by the American Geophysical Union. Paper number 93JA02929.

0148-0227/ 94/93J A-02929505.00

field componentevolvesacrossthe reversal. The paper will be structured as follows. We first give an overview of the simulation, provide details of the simulation method, and describethe initial conditions in more depth. We then presentand discussthe results of two typical simulations. Finally, we summarizeour key results.

17,391

17,392

RICHARDSON

2. Simulation 2.1.

AND CHAPMAN:

SIMULATIONS

Details

OF RELAXING

REVERSAL

d =

+ v? x n)

(2)

0B ot

(a) (4) (s)

1

VxE-

Overview

•7 x B =/•oJ

An overview of the simulation geometry is given in

Figuresla (initial configuration)and lb (subsequent evolution).The one-dimensional hybridcodeemployed hereallowsvariationin onedirectiononly (GSE z) but

V.J=0

E--V ixB-• JxB vP•+r/j

retains all three vector components. The field reversal where the electron inertial term has been neglected, is initially out of equilibrium, in that the J x B force and quasineutrality impliesnV = niV i + neVe and is not balanced by the initially isotropic ion distribuJ ne(V i -Ve). The number density n, andthebulk tion. The field lines therefore relax as time increases, ionvelocityV i areobtaineddirectlyasmoments from acceleratingthe plasma. The systemis thereforeanalogousto the evolution of a reversalwhen the associated plasmapopulationhas (by someunstatedmechanism) tional particles that represent the ions. In this one- dibeen newly isotropized. The relaxation of the reversal mensionalgeometrythe assumptionof quasi-neutrality,

thepositions andvelocities (thev•) ofthecomputa-

in the simulation occurs by some combination of fast, slow, and Alfvdn waves which can propagate in GSE +z as the field lines in principle contract in GSE+x as shownin Figure lb. This generatesa strongfield region with infiowing isotropicplasma outsideof the reversal, and a weak field region in the reversal within which plasma is accelerated, and within which also particles can become trapped. 2.2.

Simulation

Model

and continuityof charge,imply that Jz -0 (equation 5), while V.B = 0 constrainsBz to be constant.Full details of the numerical schemeare given by Terasawa

et al. [1986],with the exceptionthat in orderto obtain the electron pressurePe, Terasawa et al. assumea polytropic equationof state for the electronfluid whereaswe obtain Pe from the masslessfluid energyequation:

Ot

The simulations are performed using a one- dimensional hybrid code, in which the electron population constitutes a massless,charge-neutralizingfluid back-

+ Ve. VPe= -7P•V. Ve + (7 - 1)r/J2

(7)

where7 is the ratio of specificheats. Equations(1)(7) form a closedset, with sevenunknowns(E, B,

n, Vi, Ve, J, and P•) andsevenequations.We se-

groundand the ions(protons)are represented by com- lected the code originally developedby Terasawaet al. putational particles. The simulation grid size and time [1986]becauseit is in principlenonresistive, that is, in step are such that the ion gyromotionis fully resolved. the region of the evolving field reversal the resistivity

In this limit the full range of "low-frequency" (i.e., r/= 0 (although,of course,thereis numericalresistivity w t0

t=t0

the reversal).Consideran ionenteringthe reversalwith velocity components-via,i: and -viz ;• in the simulation

restframe(positiona). We transforminto the fieldline rest frame, where the constant convectionelectric field

Ey is zero, by movingwith velocityVTi:. The ion's x velocitycomponent becomes -(via,+ VT)i: (positiona'). The ion then travels acrossthe zero-B• region of field

line (positionsb, b') and exitsthe reversal.In the field line frame the ion's energy is constant,so its x compo-

nentsimplyreverses, becoming (via,+ VT)i:(positionc'). When

transformed

back into the simulation

rest frame

this becomes(vi• + 2VT)& (positionc). Thus the ion has gained twice the field line speedin the simulation (b)

rest frame,i.e., 2VA,a,,in the x direction.

Figure 6. (a) Potentialwelland(b) fieldlineenvisaged We should also consider the effect of the expansion for a discontinuous reversal.

of the reversal structure

in z.

The boundaries

of the

RICHARDSON

AND CHAPMAN:

SIMULATIONS

OF RELAXING

REVERSAL

17,397

regionof zeroB• eachhavevelocityV•t,zalongz. When an ion interacts with one of these boundaries, it will lose energy. A similar frame transformation argument to that given previously shows that the ion will lose twice the velocity of the boundary with eachinteraction. Hence whilst the ion approximately gains field-aligned

velocity(i.e., x directedvelocity)dueto relaxationof the reversalin x, it will losefield-aligned velocity(z directedvelocity)whilstwithinthe reversalat positions b, b'.

4.2. "First-order"

Description -11250.00

We improve on the above zero-order description by consideringthe more realistic case where B• reverses smoothly over a finite distance at time zero. The po-

.............................

0.0

5745.6

114•1.2

(0.0)

(31.7)

(63.4)

z&m(r•)

Figure •. Evolutionof the potentialwell in •he case tentiM well in this caseis harmonic(Figure8a). The B• - 2nT. The well widens and deepensin time.

potential wells observed in the simulations have this

form (Figures3, 4 and 9). Evolutionof the field line between two times to and t x is sketchedin Figure 8b. The potential well in this model is harmonic, and as Two wave structures again bound the reversal region, but these are no longer confinedto an infinitely thin the field line relaxes, the potential well expands. As an ion moves into the reversal, it interacts with the region as in the zero-orderapproximation. In this first-order model, the electric field component expandingwell and losesenergye at a rate given by

Ey acrossthe reversalis not constantbut varieswith

de dq3 z (this is mostpronounced in the simulations at early d-7 dz •o•,. (14) times).SinceBz isconstantin thisgeometry, thereisno singlede Hoffman-Tellerframetransformation velocity For suitably small initial velocitiesvi,, this lossof enVT = V• that wouldsatisfyEy(z) =-V•B, = 0, and ergy will be sufficientto renderthe ion in a boundstate. so no singleframe transformation will take us into the Thus the ion becomes tra.ppedin the reversalstructure

field line rest frame. Instead, VT is a function of z

and oscillates acrossit. Under the approximation that

and the abovefirst-orderFermiacceleration arguments motion in the x and z directions are decoupled, the cannot be employedexactly. We would expect net x- trapped ion obeysthe equation of motion directedion acceleration to be of similarmagnitudeto that deducedfrom the zero-ordermodel,that is, gains d2z E,q of lessthanor of order2V•t,•,but cannotpredictdetails = --, (15) dt 2

of the trajectories.

m

whereE, = -Oc)/Oz. Suchan equationof motion can be solvedby taking a suitablemodelfor the potential qb. A relatively simple model consistsof a simple harmonic potential well of constant depth qb0expandingfrom an initial half width A0 with constant velocity v; that is, •

(a) t=tl>t0

t=t0

z

t)-

z2 +vt)-1].

(16)

This potential arisesfrom a B• which reverseslinearly

with z; in the simulation,B• ~ tanh(z/Ao),sothispotential results when z