Resonant electron scattering caused by Alfvén wavesRodica Ciurea-Borcia, Margareta Ignat, and Ion-Dan Borcia Citation: Physics of Plasmas (1994-present) 8, 266 (2001); doi: 10.1063/1.1324990 View online: http://dx.doi.org/10.1063/1.1324990 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/8/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Self-consistent coupling of chemical, electron and radiation models for shock wave in Jupiter atmosphere AIP Conf. Proc. 1501, 1400 (2012); 10.1063/1.4769703 Energy dissipation via electron energization in standing shear Alfvén waves Phys. Plasmas 14, 062904 (2007); 10.1063/1.2744226 Nonlinear dust drift Alfvén waves in rotating planetary magnetospheres Phys. Plasmas 13, 102901 (2006); 10.1063/1.2357049 Relativistic electron beam acceleration by Compton scattering of extraordinary waves Phys. Plasmas 13, 053102 (2006); 10.1063/1.2197844 Energies of ultrarelativistic electrons produced by an oblique shock wave Phys. Plasmas 7, 4004 (2000); 10.1063/1.1290482

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Resonant electron scattering caused by Alfve ´n wavesRodica Ciurea-Borcia, Margareta Ignat, and Ion-Dan Borcia‘‘AL. I. CUZA’’ University, Faculty of Physics, 6600-IASI, Romania

~Received 7 June 2000; accepted 20 September 2000!

Pitch-angle scattering phenomenon caused by resonant interactions between relativistic electrons, 5MeV,E,15 MeV, and right circularly polarized Alfve´n waves, in parallel propagation, isexamined for the inner Jovian magnetosphere. A steady-state situation is analyzed, assuming thathot electrons are injected by a constant source, which continuously compensates for the losses at themagnetic mirrors. A parametrical numerical study is performed for different electron energies anddifferent values of electron source intensities. These theoretical results will be compared withexperimental data. ©2001 American Institute of Physics.@DOI: 10.1063/1.1324990#

I. INTRODUCTION

Working in the framework of kinetic theory, two distinctphenomena caused by nonlinear Alfve´n waves can be exam-ined: the particle acceleration and the pitch-angle scatteringdue to resonant wave–particle interactions. The first phe-nomenon, studied in Refs. 1–4, was suggested as a mecha-nism for particle acceleration in planetary magnetospheres,stellar atmospheres, extragalactic jets, or peculiar galaxies.

The electron pitch-angle diffusion was frequently de-scribed for resonant interactions between whistler waves andelectron populations with energies ranging fromE'100keV5–8 to E>1 MeV.9 This problem has large applicationsin the Earth’s radiation belts during strong magnetic stormsor artificial events occurring on a time scale greater than theparticle lifetime.

In the present article, we extend the theoretical resultsgiven by one of the authors in a previous work9 to electronpitch-angle scattering due to resonant interactions with Al-fven waves. These waves are of very low frequencies andtherefore they resonantly interact with electrons of very highenergies. Thus, we will take into consideration electronpopulations of energies 5 MeV,E,15 MeV. This situationis realistic for the inner Jovian magnetosphere (D,20RJ, RJ

is the Jovian radius,RJ>71 372 km!, where there are experi-mentally observed electrons of energies about 0.2 MeV,E,40 MeV.10–13

A steady-state situation is analyzed, modeling the Jovianmagnetosphere as a cold plasma where hot electrons comingfrom magnetic storms or other natural events areinjected.14,15We assume the existence of a constant source ofhot electrons that continuously compensates for the lossescaused by particle diffuison into the loss cone. These hotelectrons resonantly interact with electromagnetic wave per-turbations in parallel propagation, thus leading to electronscattering phenomenon. In Sec. II of this article we shalldescribe the theoretical framework from which one computesthe equilibrium wave magnetic field excited by wave–particle interactions, the stationary electron distribution func-tion established between the magnetic mirrors, the electronlifetime, and the trapped electron flux in the Jovian radiationbelts. In Sec. III a parametrical numerical study is performed

for different electron energies and different values of elec-tron source intensities. Our principal conclusions are summa-rized in Sec. IV.

II. THEORETICAL FRAMEWORK

Starting from the kinetic plasma theory, working inquasi-linear theory, one can obtain the quasi-linear diffusionequation for the relativistic case which, in stationary condi-tions, takes the following form:9

e2

2m02•

1

G sin a•

]

]a S sin a

G•

^uBf u2&

unR2ngung

] f 0~st!

]aD

1S~a,p!2P~a,p!50, ~1!

where^uBf u2& is the spatial averaged spectral density per unitfrequency,e is the electronic charge,m0 is the rest mass ofelectron, f 0

~st! is the stationary electron distribution functionnormalized to (m0c)23 (c is the light velocity!, a is thepitch-angle~formed between the electron momentum,pW , andthe magnetic field,BW 0), nR is the resonant value of electronparallel velocity,

nR5GRv2vce

GRk~2!

(vce is the electron gyrofrequency,GR5A12nR2/c2), ng is

the group velocity,S(a,p) is the spatial averaged particlesource term, chosen like in Ref. 9:

S51

n0

•

dn2

dtexpS 2

~pi2p0!2

ai2 D exp S 2

p'2

a'2 D ,

@ai5~2^pi2&!1/2, a'5~^p'

2 &!1/2#, ~3!

with n0 being the cold plasma concentration,dn2 /dt beingthe injection rate of hot electrons in cold plasma, andai ,a'

being two terms describing the beam temperatures after par-allel and perpendicular directions with respect to the mag-netic field direction. In the following paragraph the electronmomentum,p, will be expressed in eV. Its numerical valuerepresents the electron kinetic energy,Ec , connected by the

PHYSICS OF PLASMAS VOLUME 8, NUMBER 1 JANUARY 2001

2661070-664X/2001/8(1)/266/6/$18.00 © 2001 American Institute of Physics

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momentum, p, through the relation p5(1/c)AEc(Ec12m0c2). The loss term,P(a,p), is consid-ered to have its usual form:6,9,16

P~a,p!5H f 0st

Te

, a<a0

0, a.a0

,

wherea0 is the loss cone angle andTe represents the time ofescape of an electron starting from the equator, witha<a0 .17

Besides the equation that describes the stationary pitch-angle diffusion, the energy conservation law for stationaryconditions is considered, assuming the wave magnetic en-ergy, amplified along the magnetic field lines, is continu-ously lost at the magnetic mirrors owing to a smallreflection:6,9,16

R exp ~2gL~st!LRJ/ng!51. ~4!

In the above relationR represents the reflection coefficient atthe magnetic mirrors,L is the McIlwain parameter andgL

~st! isthe stationary linear growth rate, which can be written as14,16

gL~st!}h rel~Arel2Ac!5const,

with h rel the electron fraction of total electrons which par-ticipate at resonant wave-particle interactions,Arel a measureof pitch-angle anisotropy of resonant electrons, andAc thecritical anisotropy~the minimum value of the anisotropy re-quired for instability!.

In Ref. 9, from the relations~1!–~4!, there are derivedthe analytical expressions for the wave magnetic field,^uBf u2&, and for the electron distribution function,f 0

~st! , es-tablished in stationary conditions between the magnetic mir-rors. With F0

~st! known, two parameters of experimental in-terest can be computed: the particle lifetime,TL , and thetrapped electron flux in the Jovian radiation belts,JT .

III. NUMERICAL RESULTS

Using the analytical formulas computed in Ref. 9, in thepresent section we have performed a numerical study forbackground plasma parameters typical forL520 of the Jo-vian magnetosphere,18 for three values of electron energy:

~i! p055 MeV, Ti5T'5T55 MeV;~ii ! p0510 MeV, Ti5T'5T510 MeV; and~iii ! p0515 MeV, Ti5T'5T515 MeV;

and for three values of the source intensities:

~a!1

n0

dn2

dt>10217 s21, which corresponds to

dn2

dt>1028 m23 s21;

~b!1

n0

dn2

dt>10213 s21→ dn2

dt>1024 m23 s21;

~c!1

n0

dn2

dt>10212 s21→ dn2

dt>1023 m23 s21.

The reflection coefficient at the magnetic mirrors is assumedto beR'0.1 and the loss cone anglea0>0.01 rad.17

Figure 1 plots the stationary spectral density per unitfrequency, uBf u2&, versusv/vce . For each case@panels~a!,~b!, or ~c!# we have shown three curves corresponding tothree hot electrons energies@~i!, ~ii !, ~iii !#.

One can compare Fig. 4 from Ref. 6 with Fig. 1 fromRef. 9 and Fig. 1 from the present article. We notice that, for

FIG. 1. Wave spectrauBf u2& versusv/vce for different values of hot elec-tron energy and different source intensities (f ce513.5 kHz,vpe /vce520,LRJ51.53109 m, R50.1, a050.01 rad!.

267Phys. Plasmas, Vol. 8, No. 1, January 2001 Resonant electron scattering caused by Alfven waves

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electron energiesE<100 KeV, the equilibrium wave spec-trum is centered onv/vce'1022. ForE'1 MeV, ^uBf u2& isset in low frequencies range, 1024,v/vce,1022 with amaximum onv/vce'531024, while for E'10 MeV thewave spectrum ‘‘arrives’’ in the Alfve´n waves domain, cen-tered onv/vce'531025 (vci /vce55.4531024, vci is theion gyrofrequency!.

According to the increase of hot electron energies, theequilibrium wave spectra become stronger because the en-ergy involved in resonant electron–wave interactions in-creases and ‘‘moves’’ in the low-frequency range, becausehigh-energy electrons resonantly interact with low-frequencywaves. We will explain why the electrons with high energyresonantly interact with low-frequency waves. It is wellknown that the resonant wave–particle couplings are realizedfor electrons that in their own referential (n i5nR) turn withthe same angular velocity as the wave electric field~theseelectrons and the wave electric field turn together!. Becausev,vce /G, as one can observe from the relation~2!, theresonant value of the parallel velocity,nR , is negative. Thus,the electrons that participate at resonant wave–particle inter-actions and the wave perturbations move in the opposite di-rections. As the electron energy increases,upiu grows, too,and, consequently, as we can see from the relation~2!, theseelectrons will resonantly interact with electromagnetic wavesof lower frequency.

From Fig. 1, one can also notice how the Alfve´n wavespectra become narrower when the electron energy or thesource intensity increases. This aspect is determined by thediminution of the pitch-angle anisotropy with the increase ofhot electron energy or of the source intensity,dn2 /dt, espe-cially in the high-frequency part of the spectra~as we can seefrom Fig. 2!. In stationary conditions, we havegL

}hrel(Arel2Ac)5const. An increase in the source intensityor in the hot electron energy determines an increase inh rel ,too. That leads to a decrease in the system anisotropy,Arel ,for the steady-state situation.

As in Ref. 9, the anisotropy curves displayed in Fig. 2show an interesting aspect: When the source intensity in-creases, the curves corresponding toArel come closer to theAc curve. This last theoretical result has to be compared withexperimental studies performed using GEOS satellite datademonstrating that, in the wave frequency range, one obtainsArel>Ac and Arel→Ac when the intensity of the sourceincreases.7,8

The logarithm of the electron distribution function,log f 0

~st! , is plotted in momentum phase space for the threementioned cases~a!, ~b!, and ~c! with p0510 MeV, Ti

5T'5T510 MeV ~Fig. 3!. The particles injected in theJovian radiation belts are diffused due to resonant wave–particle interactions. The electron fraction that participates inthe resonant interactions increases with the increase in thesource intensity and, consequently, the diffusion process be-comes stronger. This fact determines major modifications ofthe stationary electron distribution function in comparisonwith the initial distribution of the source particles. Thus, for(1/n0) (dn2 /dt)>10217 s21, the pitch-angle diffusion phe-nomenon is weak and the equilibrium distribution form, il-lustrated by Fig. 3~a!, still keeps the ‘‘beam’’ characteristic

of the source term, given by~3!. When the injection rate ofhot electrons increases, the electrons tend to become uni-formly distributed in the phase space and their distributionbecomes isotropic, centered on 10 MeV, as one can observefrom Figs. 3~b! and 3~c!.

The trapped electron fluxJT(E1,E,E2) was estimatedusing the relation~27! from Ref. 9, forE15E0 /A2 andE2

5E0A2 with E055 MeV, E0510 MeV, andE0515 MeV.These values of electron energiesE1 and E2 correspond tothe main part of the wave spectra, for which the resonantwave–particle interactions are important. Figure 4 plots thetrapped electron flux,JT versus the injection rate of hot elec-

FIG. 2. Pitch-angle anisotropy,Arel , and critical anisotropy,Ac , versusv/vce for different electron energies and different values of the injectionrate of hot electrons (f ce513.5 kHz,vpe /vce520, LRJ51.53109 m, R50.1, a050.01 rad!.

268 Phys. Plasmas, Vol. 8, No. 1, January 2001 Ciurea-Borcia, Ignat, and Borcia

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trons, dn2 /dt, for different electron energies of the sourceterm. For low and very low injection rates of hot electrons,the loss cone is empty. As long as the loss cone is empty,when the source intensity grows,f 0

~st! changes its form, but itstays the same in terms of magnitude. Consequently,JT re-mains approximately constant,JT>1012 m22 s21>const, avalue which is in agreement with the experimental observa-tions obtained by Simpsonet al.10 or Franket al.13 When theloss cone starts to fill it, the trapped electron flux grows withdn2 /dt. This dependence upon the increase in the sourceintensity is sensitive to the ‘‘electron route’’ between themagnetic mirrors~and to the loss cone,a0 , too!, but is notmuch influenced by the electron energies. For 4 Earth’s radii(a0>0.05 rad!, JT starts to increase beginning withdn2 /dt>(10– 102) m23 s21, both for nonrelativistic elec-

trons (E>100 keV! and relativistic ones (E>1 MeV! ~see,for example, Refs. 6 and 9!. For 20 Jovian radii,JT begins togrow at very low values for the source intensity,dn2 /dt>(1025– 1024) m23 s21 ~as one can observe from Fig. 4!because, for this situation, the loss cone is very little (a0

>0.01 rad!.For the three energy values considered so far@~i!, ~ii !,

~iii !#, the particle lifetimesTL are large,TL.105 sec ~seeFig. 5!, values which correspond toTL values estimated forEarth’s magnetosphere for great values of McIlwainparameters.19 In order to ensure the stationary regime con-sidered in this article, the time interval for which the hotelectrons are continuously provided by the source term sur-passes the electron lifetime trapped in the radiation belts.Thus, the discussion from this section is applicable for elec-

FIG. 3. Phase space representation for stationary electron distribution function,f 0~st! when ~a! (1/n0) (dn2 /dt)'10217 s21; ~b! (1/n0) (dn2 /dt)

'10213 s21; ~c! (1/n0) (dn2 /dt)'10212 s21 ( f ce513.5 kHz, vpe /vce520, LRJ51.53109 m, p0510 MeV, Ti5T'5T510 MeV, R50.1,a050.01 rad!.

269Phys. Plasmas, Vol. 8, No. 1, January 2001 Resonant electron scattering caused by Alfven waves

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tron precipitation in the Jovian magnetosphere due to mag-netic storms or other natural events occurring during a timeinterval which surpasses the electron lifetime in the Jovianradiation belts (t.105 s!.

The results from this section are very sensitive to theresonant electron energy, but the form of the term source,actually, does not influence these results. Changing the termsource given by~3! with

S51

n0

dn2

dtexp X2S p2p0

aiD 2C,

for example, and keeping the electron energy domain con-stant ~and therefore the same electron momentum domain,too!, one obtains the values for the trapped electron flux,JT ,and for the electron lifetimes,TL , of the same order of mag-nitude, like in this paragraph. Only the stationary distributionfunction represented for very slow injection rates remindsone of the initial form of the term source. Asdn2 /dt in-creases,f 0

~st! becomes of the same form as those indicated byFigs. 3~b! and 3~c!.

IV. CONCLUSIONS

In this article, we have performed a parametrical studyof quasi-linear pitch-angle scattering resulting from resonantinteractions between parallel Alfve´n waves and relativisticelectrons with energies 5 MeV,E,15 MeV, in the innerJovian magnetosphere atL'20. The hot electrons are ‘‘in-jected’’ by a constant source that continuously compensatesfor the particle losses at the magnetic mirrors.

The numerical results show us that relativistic electronswith energies 5 MeV,E,15 MeV excite wave spectra inAlfven wave range, 1025,v/vce,1024. These spectra:

~1! are stronger when the electron energy or the source in-tensity grows, because the energy involved in resonantelectron–wave interactions increases;

~2! ‘‘move’’ in low-frequency range when the hot electronenergy is greater, because the high-energy electronsresonantly interact with low-frequency waves; and

~3! become narrower when the electron energy or the sourceintensity increases.

For low source intensities the equilibrium electron dis-tribution function, f 0

~st! keeps the source term characteristic.As dn2 /dt grows, f 0

~st! becomes independent of initial distri-bution of the injected electrons and tends to become isotro-pic, as one can see from Fig. 3~c!.

The trapped electron flux remains approximately con-stant, of orderJT>1012 m23 s21, until dn2 /dt'1024 andstarts to increase fordn2 /dt.1024, when the loss conestarts to fill it. JT slowly increases with the growth in hotelectron energy, but the curve shapeJT5 f (dn2 /dt) is notgreatly influenced by the particle energy.

For all the situations taken into consideration in this ar-

FIG. 4. The trapped electron flux,JT , versus the injection rate of hot elec-trons, dn2 /dt for different electron energies (f ce513.5 kHz, vpe /vce

520, LRJ51.53109 m, R50.1, a050.01 rad!.

FIG. 5. Particle lifetime,TL , versus energy for different source intensitiesand hot electron temperatures:~a! p055 MeV, Ti5T'5T55 MeV; ~b!p0510 MeV, Ti5T'5T510 MeV; and ~c! p0515 MeV, Ti5T'5T515 MeV (f ce513.5 kHz, vpe /vce520, LRJ51.53109 m, R50.1, a0

50.01 rad!.

270 Phys. Plasmas, Vol. 8, No. 1, January 2001 Ciurea-Borcia, Ignat, and Borcia

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ticle the electron lifetimes surpassTL.1025 s. Conse-quently, the discussion of this article is of interest for elec-tron precipitation in the Jovian magnetosphere due tomagnetic storms or other natural events occurring during atime interval greater than the electron lifetime in the Jovianradiation belts (t.105 s!.

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271Phys. Plasmas, Vol. 8, No. 1, January 2001 Resonant electron scattering caused by Alfven waves

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