The work considers the highly anisotropic electron distribution functions observed by Bell and co-workers at Stanford in the magnetosphere. IOt is maintained that Bells' observations provide evidence for the existence of step like deformations in electron distribution function, and that his observations may be due to such steps. The origin of such steps at the top of hiss band emissions is explained, and the implications for triggering of emissions and chorus is explained

Demekhov, AG

Kozelov, BV

Nunn, D

Pasmanik, DL

Rycroft, MJ

Titova, EE

Trakhtengerts, VY

English

Southampton (e-Prints Soton)

GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 13, PAGES 2577-2579, JULY 1, 2001Highly anisotropic distributions of energetic electronsand triggered VLF emissionsV. Y. Trakhtengerts,1 A. G. Demekhov,1 D. L. Pasmanik,1 E. E. Titova,2 B.V. Kozelov,2 D. Nunn,3 a n dM .J .R y c r o f t 4Abstract. Bell et al. [2000] have reported an exciting setof new observations, made near the equatorial plane of theplasmasphere on the L =3 :4 ﬂux tube, of hiss emissionsbetween 4.0 and 5.6 kHz, highly anisotropic energetic elec-trons, and unusual VLF triggering eects at 10.2 kHz. Herewe show that these data as a whole can be considered as therst experimental evidence of the existence of a special typeof distribution function of energetic electrons with a step-likefeature in the velocity component parallel to the magneticeld. Such a shape of velocity distribution can be crucial forexplanation of triggered VLF emissions and chorus. Assum-ing the distribution to be step-like, we can readily explainthe spatial distribution of energetic electrons along the eldline observed by Bell et al. [2000], and also strong waveamplication in the triggering process. We discuss how thestep in the velocity distribution is maintained by the hissemission. The frequency gap between the hiss band andthe triggered signal is connected with the excitation of thequasi-electrostatic mode near the upper frequency edge ofthe hiss band.1. IntroductionAn interesting highly anisotropic distribution function ofenergetic electrons (energy from 1 to 20 keV) was recentlyrevealed in POLAR particle data [Bell et al., 2000, hereafterreferred to as I], with an average equatorial pitch angle ex-ceeding  75. The measurements made along the L =3 :4magnetic ﬂux tube on 13 January, 1997, three days aftera much studied space weather event, found these energeticparticles only within 7 latitude of the magnetic equator, inaccordance with a high value of the pitch angle anisotropy.Hiss band emissions between 4.0 and 5.6 kHz, and VLF(10.2 kHz) triggered eects, were observed simultaneously(see Fig. 1 of [I]). Basing their study on these data, Bell etal. [2000] estimated the gyroresonance amplication, whichcould be sucient for VLF triggering. They considered aconventional anisotropic distribution function with a veryhigh degree of transverse anisotropy and found that the nec-essary amplication could be achieved if the higher energy(> 20 keV) electron ﬂuxes are suciently intense.On the other hand, strong amplication of an initial sig-nal, which is necessary for triggering eects, can be suppliedby a very dierent type of electron distribution function with1Institute of Applied Physics RAS, Nizhny Novgorod, Russia.2Polar Geophysical Institute RAS, Apatity, Russia.3Southampton University, Southampton, UK.4De Montfort University, Leicester, UK.Copyright 2001 by the American Geophysical Union.Paper number 2001GL013028.0094-8276/01/2001GL013028$05.00a step-like deformation in the magnetic eld aligned velocitycomponent. Such a distribution can have a moderate inte-gral anisotropy, but a very sharp local gradient in velocityspace providing strong amplication of waves in a narrowfrequency band centered on the frequency of cyclotron res-onance with the step. Step-like deformations can appearin electron distribution functions in a natural way in thecourse of development of the cyclotron instability related tothe generation of hiss. Such a deformation was actually ob-tained in calculations performed by Lyons et al. [1972], whoconsidered the formation of the energetic electron distribu-tion function under the action of plasmaspheric ELF hissemissions in the quasilinear approach. They also revealedsome evidence for the step-like feature in data from theOGO-5 satellite. Within the framework of the self-consistentquasilinear theory, a transient step-like deformation of thedistribution function of energetic electrons was obtained byTrakhtengerts et al. [1996] in computer simulations of thecyclotron instability in the Earth's magnetosphere. Distri-butions with a step are important in relation to anotherimportant problem, namely, chorus generation. As shownby Trakhtengerts [1995; 1999], a step-like deformation cangive rise to an absolute instability (backward wave genera-tion regime) that can be the basis for a quantitative theoryof chorus formation (see, as well, Villal on and Burke [1997]).The aim of this paper is to show that the results of [I] canactually serve as the rst convincing, though indirect, exper-imental evidence of the existence of step-like distributions.We demonstrate that the detected properties of the energeticelectron distribution and the strong amplication of the ini-tial wave, responsible for triggering, can be explained by thestep-like deformation without additional assumptions. Wealso discuss how the simultaneously observed VLF hiss canmaintain this deformation in the distribution function.2. Step-like distribution function andrelated eectsThe step, which appears during the development of thecyclotron instability [Lyons et al., 1972; Trakhtengerts et al.,1996; Trakhtengerts and Rycroft, 2000], divides the velocityphase space into two parts, one (with lower values of F)c o r -responding to electrons which are at some point in cyclotronresonance with the hiss band generated, and the other cor-responding to always nonresonant electrons (Fig. 1). In thecase when the value of the velocity, V?, corresponding to thestep, is much less than the characteristic velocity V0 of theenergetic electrons, we have a highly anisotropic distribu-tion. Fig. 2 shows the dependence of the energetic electronnumber density on the magnetic latitude from the equa-tor, along the L =3 :4 magnetic ﬂux tube, in the case of astep-like distribution. The case (V?=V0)=0 :2 correspondsto the observations made on POLAR. If we put the energy25772578 TRAKHTENGERTS ET AL.: ANISOTROPIC ELECTRONS AND VLF EMISSIONSFVk=V00.21V ?=V012Figure 1. Qualitative three-dimensional representation of theequatorial distribution function F, with a step separating reso-nant and nonresonant electrons.W? = mV2? =2  1:2 keV, this corresponds to the character-istic energy W0  30 keV.For whistler-mode wave generation with wave vector ~ kwalong the geomagnetic eld ~ B, the modulus of the velocityV? is related to the upper frequency !h of the hiss emissionby the cyclotron resonance conditionV? =!BL − !hkw(1)where kw = !pL!1=2h =c(!BL −!h)1=2,a n d!BL and !pL arethe electron gyrofrequency and plasma frequency, respec-tively, on the equator at L =3 :4. For a dipolar geomagneticeld, at L =3 :4, !BL=2  22 kHz.According to the experimental data [I], hiss occurs withan upper frequency limit of !h=2  5:6 kHz. The VLFtriggered signal observed simultaneously, with a higher fre-quency ftrig = !trig=2 =1 0 :2 kHz can be explained by cy-clotron amplication by a step in the distribution function(see below), if we take into account resonant cone waves witha frequency close to the resonant frequency !  !BL cos,where  is the wave normal angle between ~ k and ~ B. Underreal magnetospheric conditions the cyclotron instability gen-erates whistler mode waves inside ducts of enhanced plasmadensity, which are formed just inside the plasmapause (orby laments outside with much enhanced cold plasma den-sity). These ducts permit multi-hop propagation and theamplication of whistler waves in limited frequency bands.As the whistler-mode wave frequency ! increases, thegeneration of oblique and resonant cone waves becomesimportant. These waves are detected in satellite data[Hayakawa et al., 1986; Hayakawa and Sazhin, 1992],[Hayakawa, 1993], and their generation has been consideredin many papers [e.g., Brinca, 1972; Cuperman and Sternlier,1974; Trakhtengerts, 1963]. Our estimations in the specialcase with a step-like distribution function show a more eec-tive generation of resonance cone waves in comparison withwhistler-mode waves with ~ k k ~ B, for frequencies  0:3fBL.Then the wave frequency is not actually changing with adecrease of V?, and the cyclotron resonance condition is stillkept due to the growth of kz of the resonance cone waves.T h ew a v en u m b e ro ft h e s ew a v e si sg i v e nb ykR =!pL!1=2c(!BL cos − !)1=2 (2)with cos  !=!BL.We can now estimate when the same electrons will bein cyclotron resonance both with the resonance cone waveat a frequency of !h a n dw i t ht h ew h i s t l e r - m o d ew a v ea tafrequency of !trig. For that condition we writeV? =!BL − !hkRz(!h)=!BL − !trigkw(!trig)(3)Putting (!h=2)=5 :6k H z ,( !trig=2)=1 0 :2k H za n dL =3:4, values which follow from [I], we nd that equation (3) isfullled under the condition that cos − !=!BL  0:1, (i.e.  70).Now we estimate the one-hop cyclotron amplicationΓ of an external whistler-mode signal, which is in reso-nance with the step (see equation (3)) for a certain L-shell.For the modied Maxwellian distribution function F V2j? exp(−V2=V20 )(V? − VkL), where j is the anisotropyindex,  is the Heaviside unit function and VkL is electronparallel velocity at the equator, Γ can be evaluated asΓstep  2(ka)1=3 Γ0; Γ0 ' 0:3nhL ancL V0!BL (4)[Trakhtengerts et al., 1996; Hobara et al., 1998], where Γ0 isthe amplication for the standard distribution function (i.e.,with (V? −VkL) replaced by unity, the numerical factor inΓ0 being weakly dependent on the anisotropy index j), nhLand ncL are the hot and cold electron densities, respectively,and a = LR0=p3 is the characteristic length scale of theEarth's magnetic eld near the equator. The amplicationΓ for the smooth anisotropic distribution function dependsonly weakly on the anisotropy index for high anisotropy [Bellet al., 2000] and is close to Γ0. Putting, in accordance with[I], nh(W<20 keV)  10−2 cm−3, V0  8  109 cm s−1,ncL  2:5  102 cm−3, k  3k m−1,a n da  1:28  104 km(for L =3 :4), we nd that Γ0  0:24 and Γstep  15, or60 dB. It is enough to explain the observed triggered eect.0 5 10 15 20latitude00.20.40.60.81n / nL123Figure 2. The dependence of the energetic (hot) electron den-sity as a function of latitude (in degrees) along the magnetic ﬂuxtube in the case of a step-like distribution for: 1 { V?=V0 =0 :01,case 2 { V?=V0 =0 :05, case 3 { V?=V0 =0 :2. Here nL is theenergetic (hot) electron density in the equatorial plane.TRAKHTENGERTS ET AL.: ANISOTROPIC ELECTRONS AND VLF EMISSIONS 25793. DiscussionThe above analysis demonstrates an important feature ofwave-particle interactions, namely, their strong dependenceon the ne structure of the distribution function F of theenergetic particles. In particular, a step-like deformation ofF in the velocity component Vk yields the cyclotron ampli-cation Γstep which is much larger than the amplication Γ0by a smooth anisotropic distribution function (according tothe above estimates, Γstep=Γ0  60). The step-like defor-mation can play a crucial role for the generation of discreteELF/VLF emissions [Trakhtengerts, 1995, 1999]. Experi-mental detection of such a feature is rather dicult becauseit requires an instrument with a very ne resolution in Vk( 1). Moreover, as a rule this is a transient phenomenon,with a duration of less than one second. Electron data withsuch velocity and temporal resolution are not yet possible.Nevertheless, some indications of a step in the distributionfunction were revealed in OGO-5 satellite data [Lyons et al.,1972]. Unfortunately, the energy and pitch angle resolutionof these data ( 10% and  5t o1 0, respectively) werenot enough to reconstruct the exact width and dynamic fea-tures of the step. From this point of view, the complex dataset on energetic-electron distributions and whistler waves,presented in [I], can be considered to be the rst actual ev-idence for the existence of step-like distribution functions,while the observed triggering eects yield some informationabout very fast processes. Simultaneous observations of thehiss, responsible for the formation of the step-like featureprovide additional support for this suggestion.As an additional experimental check of the model dis-cussed, a comparison of the electric and magnetic eld com-ponents of the hiss band could be made. We predict thatthe electric component should become stronger towards theupper frequency limit of the hiss band. It would also beimportant to determine the direction of the wave vector ~ kas a function of frequency. According to the above model,two particular directions of ~ k must be present. It is knownthat two wave-normal directions can be present in satelliteVLF data [Hayakawa et al., 1986; Hayakawa and Sazhin,1992; Hayakawa, 1993], but it is necessary to undertake amore sophisticated data analysis, including an investigationof the dependence of the ~ k d i r e c t i o no nf r e q u e n c y .Acknowledgments. This work was supported by NATOLinkage grant ESR.CLG 975144 and INTAS grant No. 99-00502.ReferencesBell, T. F., U. S. Inan, and R. A. Helliwell, Simultaneous triggeredVLF emissions and energetic electron distributions observedon POLAR with PWI and HYDRA, Geophys. Res. Lett., 27,165{168, 2000.Brinca, A. L., On the stability of obliquely propagating whistlers,J. Geophys. Res., 77, 3495{3507, 1972.Cuperman, S., and A. Sternlier, Obliquely propagating unstablewhistler waves: A computer simulation, J. Plasma Phys., 11,175{188, 1974.Hayakawa, M., M. Parrot, and F. Lefeuvre, The wave normals ofELF hiss emissions observed onboard GEOS 1 at the equatorialand o-equatorial regions of the plasmasphere, J. Geophys.Res., 91, 7989{7999, 1986.Hayakawa, M., and S. S. Sazhin, Mid-latitude and plasmaspherichiss: A review, Planet. Space Sci., 40, 10, 1325{1338, 1992.Hayakawa, M., Study of generation mechanisms of magne-tosphric VLF/ELF emissions based on the direction ndings,in Proceedings of the NIPR Symposium on Upper AtmospherePhysics, 6, 117{133, 1993.Hobara, Y., V. Y. Trakhtengerts, A. G. Demekhov, andM. Hayakawa, Cyclotron amplication of whistler waves byelectron beams in an inhomogeneous magnetic eld, J. Geo-phys. Res., 103, 20,449{20,458, 1998.L y o n s ,L .R . ,R .M .T h o r n e ,a n dC .F .K e n n e l ,P i t c ha n g l ed i  u -sion of radiation belt electrons within the plasmapause, J. Geo-phys. Res., 77, 3455{3474, 1972.Trakhtengerts, V. Y., On generation mechanism of VLF emissionsin external Earth's radiation belts, Geomagn. Aeron., 3, 442{451, 1963.Trakhtengerts, V. Y., Magnetosphere cyclotron maser: Back-ward wave oscillator generation regime, J. Geophys. Res., 100,17,205{17,210, 1995.Trakhtengerts, V. Y., A generation mechanism for chorus emis-sion, Ann. Geophys., 17, 95{100, 1999.T r a k h t e n g e r t s ,V .Y . ,M .J .R y c r o f t ,a n dA .G .D e m e k h o v ,I n t e r -relation of noise-like and discrete ELF/VLF emissions gener-ated by cyclotron interactions, J. Geophys. Res., 101, 13,293{13,303, 1996.Trakhtengerts, V. Y., and M. J. Rycroft, Whistler-electron in-teractions in the magnetosphere: new results and novel ap-proaches, J. Atmos. Solar-Terr. Phys., 62, 1719{1733, 2000.Villal on, E., and W. J., Burke, Theory of quasi-monochromaticwhistler wave generation in the inner plasmasphere, J. Geo-phys. Res., 102, 14,381{14,395, 1997.V.Y.Trakhtengerts, A.G.Demekhov, and D.L.Pasmanik, In-stitute of Applied Physics RAS, 46 Ulyanov Street, 603600Nizhny Novgorod, Russia (e-mail: vyt@appl.sci-nnov.ru;andrei@appl.sci-nnov.ru; pdl@aurora.appl.sci-nnov.ru)E. E. Titova and B. V. Kozelov, Polar Geophysical InstituteRAS, 184200 Apatity, Russia (e-mail: titova@pgi.kolasc.net.ru;kozelov@pgi.kolasc.net.ru)D. Nunn, Department of Electronics and Computer Science,Southampton University, Higheld, Southampton, SO17 1BJ, UK(e-mail: dn@ecs.soton.ac.uk)M. J. Rycroft, Faculty of Computer Sciences and Engineer-ing, De Montfort University, Leicester, LE 1 9BH, UK (e-mail:Michael.J.Rycroft@ukgateway.net)(Received February 17, 2001; accepted March 21, 2001.)

A generation mechanism for chorus emission,

Cyclotron ampli of whistler waves by electron beams in an inhomogeneous magnetic

Institute of Applied Physics RAS, 46 Ulyanov Street, 603600 Nizhny Novgorod, Russia (e-mail: vyt@appl.sci-nnov.ru; andrei@appl.sci-nnov.ru; pdl@aurora.appl.sci-nnov.ru)

L y o n s ,L .R . ,R .M .T h o r n e ,a n dC .F .K e n n e l ,P i t c ha n g l ed i  u -sion of radiation belt electrons within the plasmapause,

Magnetosphere cyclotron maser: Backward wave oscillator generation regime,

Mid-latitude and plasmaspheric hiss: A review,

Obliquely propagating unstable whistler waves: A computer simulation,

On generation mechanism of VLF emissions in external Earth's radiation belts,

On the stability of obliquely propagating whistlers,

Simultaneous triggered VLF emissions and energetic electron distributions observed on

Study of generation mechanisms of magnetosphric VLF/ELF emissions based on the direction

T r a k h t e n g e r t s ,V .Y . ,M .J .R y c r o f t ,a n dA .G .D e m e k h o v ,I n t e r -relation of noise-like and discrete ELF/VLF emissions generated by cyclotron interactions,

The wave normals of ELF hiss emissions observed onboard GEOS 1 at the equatorial and o-equatorial regions of the plasmasphere,

Theory of quasi-monochromatic whistler wave generation in the inner plasmasphere,

Whistler-electron interactions in the magnetosphere: new results and novel approaches,

Highly anisotropic distribution of energetic electrons and triggered VLF emissions

Highly anisotropic distribution of energetic electrons and triggered VLF emissions

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