Auroral kilometric radiation and electron pairing
MManuscript prepared for Journalnamewith version 3.1 of the L A TEX class copernicus2.cls.Date: 24 July 2020
Auroral kilometric radiation and electron pairing
R. A. Treumann and W. Baumjohann a International Space Science Institute, Bern, Switzerland Space Research Institute, Austrian Academy of Sciences, Graz, Austria Geophysics Department, Ludwig-Maximilians-University Munich, GermanyCorrespondence to: [email protected]
Abstract .– We suggest that pairing of bouncing medium-energy electrons in the auroral upward current region closeto the mirror points may play a role in driving the electroncyclotron maser instability to generate an escaping narrowband fine structure in the auroral kilometric radiation. Wetreat this mechanism in the gyrotron approximation, for sim-plicity using the extreme case of a weakly relativistic Diracdistribution instead the more realistic anisotropic Jüttner dis-tribution. Promising estimates of bandwidth, frequency driftand spatial location are given.
The plasma dynamics in the near-Earth high-altitude auroralmagnetosphere is comparably easy to monitor, either fromground or space (e.g., Paschmann et al., 2003). The pecu-liarity of this region lies in the overlap of the dense, cold,electrically neutral atmosphere and the dilute, fully ionized,collisionless, hot magnetospheric plasma which occurs ina narrow layer of roughly 100-300 km ( ∼ .
03 R E ) verti-cal extent. It forms the topside su ffi ciently electrically re-sistive ionosphere to, at its bottom, allow cross-magneticfield currents to close the magnetic field-aligned currents thatconnect the distant magnetosphere and Earth. Continuouslygrowing evidence suggests their origin in reconnection in thecentral-magnetotail cross-field current-sheet. From a terres-trial viewpoint the currents flow either upward or downward.Depending on the wave or particle picture, they are carriedby (kinetic) Alfvén waves or electrons.In the particle picture electrons of energies E e in the rangeof several keV emerge from the reconnection sites to flowdownward along the magnetic field into the ionosphere, car-rying upward currents. Correspondingly, downward currentsare carried by low-energy, E e (cid:46) few 0.1 keV, upward ac-celerated ionospheric electrons. This picture is well estab-lished and was strongly supported by observations from theViking (Lundin et al., 1987; de Feraudy et al., 1987), Freja(cf., Lundin et al., 1994), and FAST (see the special issue onFAST, introduced by Carlson et al., 1998) spacecraft. Among the processes related to field-aligned currents,generation of Auroral Kilometric Radiation (AKR) (Gur-nett, 1974) and its fine-structure still remains insu ffi cientlyunderstood. AKR is broadband and intense, propagating inthe free space magneto-ionic X mode. It releases several percent of the total auroral energy available during magneto-spheric disturbances into free space, an amount substantiallyexceeding any reasonable gyro-synchrotron radiation abovethe X or O mode cut-o ff s. AKR is non-thermal and highlyvariable. Its mechanism has been convincingly traced backto direct amplification of free space radio waves, mainly inthe X mode, by the Electron Cyclotron Maser instability(ECMI)(Sprangle & Drobot, 1977; Sprangle et al., 1977; Wu& Lee, 1979; Melrose, 1989, 2008; Cairns et al., 2005; Bing-ham et al., 2013; Speirs et al., 2005, 2008, 2014; Treumann,2006; Mcconville et al., 2008), a possibility based on nega-tive absorption (adopted from maser theory). This was firstidentified by Twiss (1958) and Schneider (1959) as a possi-bility to convert a plasma from an absorber into a radiator,while not yet referring to the electron cyclotron instability.It requires an underdense background plasma immersed intothe strong auroral magnetic field B . The ratio of cyclotron toplasma frequency ω c /ω e > ffi cient to change sign, allowing the plasma volume as awhole to radiate quasi-coherently like masers or lasers in theinfrared respectively optical bands. The currently favouredcondition resembling this kind of excitation is the electronloss-cone distribution, lacking electrons in a narrow field-aligned angular domain. They mimic the necessary excitedstate. The free energy stored in the loss cone is directly trans-ferred to the free space modes by the ECMI.Attempts of the plasma to refill the loss-cone by low fre-quency resonant wave-particle interaction with VLF (La- a r X i v : . [ phy s i c s . s p ace - ph ] J u l R. A. Treumann & W. Baumjohann: AKR and pairingBelle & Treumann, 2002) are slow quasilinear processes un-der the dilute topside conditions. They e ff ectively limit thehigh-energy radiation belt fluxes (Kennel & Petschek, 1966)but cannot come up for either depleting the lower energy au-roral field-aligned electron loss cone or explaining the vari-ability and fine structure of AKR, though models have beenput forward (Louarn et al., 1996) to overcome this deficiency.Other mechanisms have also been proposed, based onelectron-hole dynamics (Treumann et al., 2011); for theirnonlinearity they did not find general acceptance. Thoughthe existence of holes has been observationally confirmed(Ergun et al., 1998a,b; Pottelette et al., 1999; Pottelette &Treumann, 2005), and the electron hole mechanism nicelyreproduces several properties of the radiation, it lacks con-firmation. Holes are mesoscopic features few Debye lengthslong only. It is di ffi cult to see how they could e ff ectively am-plify km-lengths waves. This might still be possible statisti-cally as holes exist in very large numbers organized in chainsalong the magnetic field. They might act collectively over theAKR kilometer wavelength to amplify radiation, as has beensuggested (Treumann et al., 2011). Such a detailed stochasticcalculation is still missing.In the following we propose, at least qualitatively, anotherpromising mechanism possibly capable of causing the spatialand temporal fine structure observed in AKR. This mech-anism can be based on the resonant interaction of quasi-trapped electrons with propagating plasma waves generat-ing an attraction between electrons spaced by a Debye lengthalong the field. Field aligned electrons move in the geomagnetic mirror ge-ometry. Downward current-upward electrons of low iono-spheric energy, starting at large pitch-angles in the up-per ionosphere, when propagating outward into the mag-netosphere, conserve their magnetic moments and becomequickly completely field aligned. Upward current-downwardelectrons on the other hand, starting at small pitch-angles intheir low-field reconnection site, while moving along theirseparatrices, increase their pitch-angle. Some of them mayultimately become trapped in the magnetospheric field if onlytheir mirror points lie above the ionosphere.Reconnection in the tail current sheet provides two elec-tron populations, an almost strictly field-aligned populationescaping along the separatrix, and an exhaust component thatimpacts on the reconnection-caused plasmoids and is scat-tered into larger pitch-angles. These latter electrons remainmagnetospherically trapped along the auroral magnetic fieldwith mirror points remaining well above the ionospheric cur-rent closure.Recently we have shown that, in a magnetic mirror ge-ometry, conditions can evolve when classical electron pair-formation occurs. This process generates a very high thermal (pitch-angle) anisotropy of the paired component. Conditionsof this kind evolve in mirror modes but are also expected inthe upward current region. Below we qualitatively developthis scenario to some detail as a preparation to its quantita-tive investigation via a more elaborate analytical theory of itsproperties, or via numerical simulations.
Classical electron pairing is based on collective over-screening of the bare electron charge outside the Debyesphere of moving electrons in resonance with a plasma waveof frequency ω k . The parallel electric potential Φ caused byan electron of velocity v is obtained from Φ ( x , t ) = − e (2 π ) (cid:15) (cid:90) d ω d k δ ( ω − k · v ) k (cid:15) ( k ,ω ) e i k · ( x − v t ) (1)where the dielectric function contains the susceptibilities ξ i , e contributed by the waves and the Debye screening term( k λ D ) − . Note that the time variation of any electromagneticpotential A ( x , t ) does not contribute. Since the dielectric ap-pears in the denominator its inverse comes into play. Quitegenerally it can be written1 (cid:15) ( ω, k ) = k λ D + k λ D (cid:18) + ω k ω − ω k (cid:19) (2)where ω k is the solution of the dielectric (cid:15) ( k ,ω ) = k c /ω ,considering only the plasma eigenmodes. The potential de-pends on the resonant electron contribution with the eigen-modes.The inverse dielectric suggests that, in particular cases, thepotential can become negative, if only theplasma wave fre-quency ω k (cid:38) ω , in which case the second term in the brack-ets becomes negative and larger than one. Once this happens,the Debye term is superseded by the resonance. The contri-bution to the above electric potential Φ then is negative, suchthat Φ > -integral changes sign. Screening then results in at-traction outside the Debye radius x > λ D . This corresponds toan overscreening of the electron charge in resonance ω/ k ≈ v with the particle speed which happens in the region behindthe particle. What is thus needed for this to happen is that theelectron speed is comparable to the phase velocity v ∼ ω/ k ,i.e. resonance. .For ion-sound waves we derived the precise conditions(Treumann & Baumjohann, 2019). The physics of attractivepotential generation is the same for kinetic Alfvén waves(kAW) but the conditions are slightly di ff erent (Narita et al.,2020b). We refer to the latter paper for details even thoughthey are not as important for the purposes of the present com-munication. For the kinetic Alfvén wave dielectric one has (cid:15) kAW ( ω, k ) = + k λ D + c V A (cid:32) + k ⊥ λ e (cid:33)(cid:34) + (cid:16) k · r ci (cid:17) (cid:32) + T e T i (cid:33)(cid:35) − (3) It is interesting to note that a similar e ff ect would also resultin purely growing / damped waves where ω k = ± i γ k , an e ff ect not yetexplored anywhere. Figure 1.
Fine structure in auroral kilometric radiation FAST observations (Figure adopted from Treumann et al., 2011). Top: two narrowbands of intense and highly temporally structured drifting auroral kilometric emissions. The bandwidth is of the order of a few kHz only.Upward drift of the bands implies downward motion of the source into the stronger ambient magnetic field region. Obviously each bandconsists of many separate short emission events. Bottom: High temporal resolution of the indicated available time interval clearly showing thesuperposition of the many microscopic radiation events which make up the two radiation bands. Each event moves first upward in frequency(sources moves downward in space), then turns around and moves back upward. Emission is strongest in the turn-around (reflection). with r ci = v i ⊥ /ω ci the vectorial ion gyro-radius, and λ e = c /ω e the electron inertial length. The relevant wave fre-quency applicable to the strong field auroral region for β = µ NT e / B (cid:46) √ m e / m i is ω k A ≈ k V A / (cid:113) + k ⊥ λ e (4)with V A (cid:28) c β >
1, the other limiting case). It is long-wavelength and, in a hot plasma, propagates somewhat fasteralong the magnetic field than the ordinary Alfvén wave, andperpendicular to it roughly several ten times slower. In coldplasma where electron inertia cannot be neglected it is in-stead slower than the Alfvén wave. For our purposes here itsu ffi ces to know that the kAW propagates approximately atAlfvén speed and, in the topside AKR source slows some-what down. Here the ion contribution plays no role, and onehas approximately V A → V A / (cid:112) + k λ e . The weak wave elec-tric potential (Lysak & Lotko, 1996) resulting from its kineticnature is along the magnetic field and is believed to be re-sponsible for some electron acceleration, sometimes (in theolder literature) claimed to come up for the whole auroralelectron energy. In view of the tail-reconnection set-up thisseems improbable; the electrons start from there with energyin the right range. Any attractive pairing e ff ect in resonancewith those fast parallel electrons will be in this direction aswell but is independent on the wave field as pairing is a sin-gle particle e ff ect which causes a local change in the dielec-tric constant of the plasma but otherwise has no e ff ect onthe wave Alfvén dynamics. Here we explore its importancefor electron dynamics and generation of AKR. Naturally,like any Alfvén wave, the kinetic Alfvén wave possesses aperpendicular electric field which under certain conditionscauses the electrons to perform a perpendicular drift motionwhich displaces the electrons very slowly from their originalto a neighbouring flux tube, an e ff ect which we safely ignorebelow. Wave-electron resonance between the electron and kAW pro-duces a negative potential in the electron wake which trapsanother electron, a process that modifies the plasma dielec-tric. Attractive potentials were first suggested half a cen-tury ago by Neufeld & Ritchie (1955) who, however, didnot suggest any pairing mechanism. They considered Lang-muir waves which, however, turned out to have no impor-tance anywhere. The idea was later picked up in applicationto ion-sound waves (Nambu & Akama, 1985). In fact, in anun-prepared plasma, attractive potentials and pairing are in-deed unimportant, except of possibly justifying the assump-tion of compound formation, as is assumed in pic simulationsin plasma, Treumann & Baumjohann (see 2014).In particular cases attractive potentials may play a rolein dilute magnetized plasmas when electrons in magnetic Neufeld & Ritchie (1955) introduced it even before Cooper’sinvention of electron pairs (Cooper, 1956; Bardeen et al., 1957) ofopposite spins, the idea of which Cooper might have gotten fromthere (and another paper by Fröhlich, 1952, who did not succeed)and applied successfully to cold fermions. mirror geometries undergo bounce motions, conditions oc-curring for instance in mirror instabilities,(Treumann &Baumjohann, 2019). For the description and precise calcu-lation of the mechanism causing classical electron pairing,one may consult that paper. Pairing in plasma is based on theevolution of the attractive potential in the wake of a movingelectron and can, in principle be extended to multiple elec-trons. Classically, at the applied high temperatures the pairedelectrons have of course for long lost their fermionic prop-erty.Since the wake is of much longer scale than the Debyelength λ D = v e /ω e , with mv e = T e the classical thermal speedof the electron at temperature T e , and ω e the electron plasmafrequency, the attractive potential evolves outside the elec-tron Debye sphere. It unavoidably a ff ects many electronswhich are separated at distance N − / (cid:28) λ D , with N back-ground plasma density. Thus there will always be at least an-other electron which moves at approximately same speed andthus feels the attractive potential of the first which, howeverbecomes modified by its presence, with modification rapidlydecreasing, the more electrons participate.The condition that the trapping is about stable is thatthe parallel energy di ff erence between the two electrons is u (cid:107) < | e Φ | / m , where u (cid:107) is defined below. We have shown thatthe basic e ff ect is that two electrons bind together in the at-tractive potentials which overcompensates the remaining re-pulsive potential outside the Debye sphere. It requires thepresence of a low frequency plasma wave propagating par-allel to the two interacting electrons and is in resonance. Theresonance is a single particle e ff ect and no plasma resonance.The main condition is that the parallel phase velocity ω k / k (cid:107) of the wave and the parallel centre-of-mass velocity U (cid:107) =
12 ( v (cid:107) + v (cid:107) ) (5)of the two electrons 1 , λ D < ∆ s (cid:46) . λ D outside the Debye sphere are about thesame. This implies that for the two coupled electrons holds u (cid:107) = | v (cid:107) − v (cid:107) | (cid:28) | U (cid:107) | (6)The centre of mass velocity of the electrons parallel to thefield in the bounce motion would decrease to zero when, afterhaving started from some location s (0) along the field, theyapproaching the common mirror point s m .In paired resonance with the kAW the resonance stops thebounce motion as long as the particles remain to be paired.The pair of electrons then continues moving as a triplet to-gether with the wave at the slow wave speed ω/ k (cid:107) ≈ V A alongthe field. Thus, when the attraction comes into play, the par-allel speed of the pair drops to that of the wave.Assume that the electrons initially each had high energy E e and thus high parallel speed U (cid:107) Figure 2.
Another example of narrow band emission in auroral kilometric radiation (after Treumann et al., 2011) observed by FAST withno high-resolution observations available. In this event time resolution does not permit resolving all microscopic emission sources. Only fewspatially downward (upward in frequency) moving sources can be identified. The emission band is about stably in spatial (and frequency)location. Apparently most moving microscopic radiation sources move upward in space (downward in frequency) from their location ofstrongest emission. transferred to their perpendicular speeds v ⊥ , i.e. the gyra-tion. At short distance from s m the parallel speed U (cid:107) has de-crease so much that it equals the phase velocity ω k / k (cid:107) of thekAW, and the attractive potential evolves. The newly formedpair becomes locked to the plasma wave by the induced lo-cal change in the dielectric function which is responsible forthe attractive potential. The triplet of the two electrons andthe plasma wave then continues moving together at the slowphase speed, and the paired electrons completely drop outfrom bounce, both maintaining their perpendicular speeds.Conservation of the magnetic moments, which a ff ects onlythe perpendicular energy implies that the latter increases.Thus pairing causes adiabatic perpendicular heating of thepairs. At this point their parallel speed has become U (cid:107) ( s (cid:46) s m ) ∼ ω/ k (cid:107) (cid:28) U (cid:107) (0) (7)much less than their initial parallel velocity U (cid:107) (0).When this happens, the perpendicular individual electronspeeds equal their initial velocities v ⊥ , ( s m ) ≈ v , ( s = s is the distance along the magnetic field from the re-connection to the pairing site, which is close to the initial mir-ror point s m . Note that the trapping condition is that electrons1 and 2 have about same parallel speed, which is given by theabove condition on u (cid:107) . It fixes the parallel speed, locks theelectrons to the wave, and thus, by conservation of the elec- tron magnetic moments µ = E ⊥ ( s ) / B ( s ), transfers all their re-maining individual energies into their perpendicular speeds.Since the cyclotron frequency depends only on the magneticfield B ( s ) at that spatial location, this process modifies thegyroradii of the two electrons.Assume that the perpendicular speed is roughly abouttheir average thermal speed v e , then their perpendicular en-ergy in pairing becomes comparable to their initial energy(temperature T b ) which is the mean energy of the bouncingplasma component. It is high above that of the local ambientplasma if any. The error made in this assumption is small,because the energy spread of the fast electrons generated intail-reconnection must necessarily be small. The pitch-anglespread of the field-aligned electrons in the topside auroral re-gion is reduced to a few degrees only, filtering out a narrowrange of energy spread available for pairing. Those electronsare close to mono-energetic and possess a large energy (ther-mal) anisotropy which can roughly be approximated as A p = E ⊥ ( s m ) E (cid:107) ( s m ) − ≡ tan θ ( s m ) − ≈ E e (0) mu (cid:107) (cid:29) mu (cid:107) /
2, and in bounce motiontan θ ( s m ) = lim s → s m (cid:20) B ( s m ) B ( s ) − (cid:21) − (cid:29) u (cid:107) , this anisotropy is huge.As written here, it is an energy ratio. However, in a volumeof many Debye lengths along the magnetic field in which alarge fraction of electrons, i.e. a large fraction of electrons inthe Debye sphere contribute to trapping, each carrying alongits paired partner electron at a distance of λ D +∆ s while prop-agating down the field along with the kAW. There will be asubstantial fraction of such pairs in the flux tube over one orseveral wavelengths of the kAW. These may not be stable forvery long time due to fluctuations but new pairs will contin-uously reform within the unceasing stream of electrons sup-plied by tail reconnection. Such a volume provides a fairlylarge energetic anisotropy and, under not too restrictive con-ditions, may drive the ECMI unstable.Electrons, after acceleration and ejection from the recon-nection site (the reconnection exhaust, as it is sometimescalled) have nominal velocity v (cid:38) km / s. Their parallelspeed is v (cid:107) = v cos θ ( s ). The phase velocity of kAW is of theorder of ω/ k ∼ km / s. When the electron approaches res-onance, the pitch-angle has changed to satisfy the conditioncos θ ( s ) ∼ ω kv ∼ V A v (cid:46) − (10)which close to its mirror point s m corresponds to a pitch-angle θ ( s ) (cid:38) ◦ . Its perpendicular speed is thus practi-cally v , and the anisotropy in energy A (cid:38) , as arguedabove. If contributed by a susceptible number of particlesan anisotropy that high must have a profound e ff ect on thegeneration of radiation.Total plasma densities in the broad upward current AKRsource region amount to at most N ∼ a few times 10 m − ,corresponding to a dilute and even underdense strongly mag-netized plasma whose cyclotron frequency is around ω c / π ≈
300 kHz corresponding to a magnetic field of B (cid:46) nT( (cid:46) . λ D ∼ d ∼ − m. This givesroughly 150 particles within a length of 1 . λ D . Let the num-ber density of pairs be N p , then a probably realistic pair-to-plasma density ratio would be N p / N ∼ − or so, whichmeans that just every thousandth electron would capture an-other one to form a pair. The remaining electrons form theplasma background and do not contribute to any anisotropy.They might, however, be subject to a weak loss-cone distri-bution of the kind of a Dory-Guest-Harris (DGH) or Ashour-Abdalla-Thorne (AAT) distribution. Background and pairedelectron populations are completely independent. They cou-ple only by quasi-neutrality N = N + N p .Below we examine, for simplicity and to demonstratethe possible excitation of radiation, just one particular well-known emission model which may apply to the distributionof paired electrons. This is the gyroresonant (gyrotron) sce-nario. The simplest imaginable radiator of electromagnetic wavesin a highly anisotropic plasma is the gyrotron. Radiation isdue to electron bunching in the unstable free-space radia-tion wave field of frequency ω and wave number k , not thekAW field! This wave field obeys the electromagnetic dis-persion relation N ≡ k c /ω ≈ ffi ciently far above X-mode cut-o ff . It implies the presence of a highly anisotropicparticle distribution capable of directly amplifying one ofthe free-space magneto-ionic radiation modes. Its e ff ect is amaser emission driven by the anisotropy. It has been sug-gested originally for plasma devices, gyrotrons (Gaponov,1959; Gaponov et al., 1967). The highly anisotropic electron(pair) distribution in the parallel moving frame of the kAW isthen modelled as a gyrotropic Dirac distribution with all the(relativistic) momentum p ⊥ in the perpendicular direction f p ( p (cid:107) , p ⊥ ) = N p π p ⊥ δ ( p ⊥ − p ⊥ ) δ ( p (cid:107) − m γω k / k (cid:107) ) (11)neglecting here the small velocity spread u (cid:107) in the paralleldirection and any possible spread in the distribution of per-pendicular momentum p ⊥ due to any given initial electrondistribution, in order to make it analytically accessible. In-cluding such a spread would imply the use of an anisotropicJüttner distribution (Treumann & Baumjohann, 2016) whichcomplicates the problem substantially. For our purposes itsu ffi ces to stick to the simplest model first. In the paired statethe electrons move with wave phase velocity V A (cid:28) c alongthe magnetic field. We therefore transform to the wave frame p (cid:107) A setting δ ( p (cid:107) − mV A ) → δ ( p (cid:107) A ). [Actually, this transforma-tion is not as simple because the relativistic factor γ also de-pends on perpendicular momentum, a complication whichwe neglect here as the small Alfvén velocities .] Then, inthe laboratory frame, any displacement of the pair along thespatially changing (increasing or decreasing) magnetic field B ( s ) appears, in the observed pair-caused emission spectrum,simply as a frequency drift. The perpendicular particle mo-ment must however be treated relativistically with initial rel-ativistic factor γ = (cid:113) + p ⊥ / m c . Moreover one assumesthat k ⊥ ρ ≈ ρ . In this case, in the Bessel expansion of the plasmadielectric (cf., e.g., Baumjohann & Treumann, 1996) only theharmonics n = , ± N ≡ k c ω = − ω p ω (cid:20) ωω − ω c + p ⊥ m c k c − ω ( ω − ω c ) (cid:21) (12)The last term in the parentheses is the relativistic correctionwhich turns out to be crucial. The growing solutions of thisrelation with positive imaginary part ω i > ω i ω c ≈ √ (cid:20) p ⊥ m c ω p ω c (cid:18) − N cos θ (cid:19)(cid:21) (13)with N ≈ − N cos θ ≈ sin θ , under the condition ω p /ω c (cid:28) sin θ ( p ⊥ / mc ) (Melrose, 1989). One hence re-quires that the emission is very close to perpendicular suchthat sin θ ≈
1. One also needs a strong magnetic field B andlow density N p of the pairs to have ω p (cid:28) ω c , and the ini-tial momentum, respectively the relativistic β = v ⊥ / c of theelectrons, should not be too small. For ∼
10 keV electronsone has roughly β ≈ .
1. Emission at the fundamental in itsturn then requires ω p /ω c < − − − . This is not unrea-sonable in view of the rough order-of-magnitude discussiongiven above. Estimated densities would approximately cor-respond to this condition. One, however, expects that evenunder these conditions the presence of the non-paired plasmabackground would absorb radiation at the fundamental, suchthat the intensity of the radiation in the fundamental shouldbecome low.The present calculation has been done just for the fun-damental harmonic | n | =
1. Radiation at higher harmonics ismuch less vulnerable to absorption in the diluted plasma ofthe AKR source region. There can be no doubt that higherharmonics | n | > | n | =
1, while escaping re-absorption. Unfortunately inclusion ofhigher harmonics becomes substantially more complicated(Melrose, 1989) analytically because all the di ff erent Besselfunctions of higher order | n | > ff ect would become. With inclusion of low higherharmonics, the relativistic resonance condition (frequencymismatch) reads ∆ n = ω − n ω c /γ − k (cid:107) v (cid:107) (14)The harmonic number just factorizes the relativistic cy-clotron frequency. In order to infer the e ff ect of | n | > n ω c to obtain that ω i , n n ω c ∼ (cid:20) p ⊥ m c ω p n ω c (cid:18) − N cos θ (cid:19)(cid:21) (15)One therefore expects the growth rate of the low higher har-monics weakly decreases as ∼ n − / when normalised to its n th cyclotron harmonic, which implies that at the lowest har-monics the growth increases with respect to the fundamentalincreases as ∼ n / . In addition to the fundamental with its problem of escaping, reabsorption, and quasilinear quench-ing when remaining trapped, those low harmonics may in-deed dominate the observation. Realizing this fact is impor-tant as it a ff ects the interpretation concerning the involvedmagnetic fields.The above result, though still quite imprecise, can also beinterpreted as n − sin θ which means that, holding up the con-ditions on the density, the emission at higher harmonics ismore oblique . In addition (see Melrose, 1989) the frequencymismatch ∆ n must be positive in order to escape from theplasma. This is anyway necessary but easier to satisfy forhigher harmonics than the fundamental. ∆ n > n th harmonic, and the condition on theparameters becomes ω p / n ω c (cid:28) β sin θ which is slightly lessrestrictive on the density, while relaxing the escape conditionfor all harmonics under consideration which are above the X-mode cut-o ff . Thus the expectation is that one observes pair-excited higher harmonics rather than fundamental radiationas the latter will be suppressed. Above we tentatively applied the idea of electron pair for-mation due to the generation of attractive potentials betweenelectrons in bounce resonance with kinetic Alfvén waves inthe auroral region. In the simplest gyrotron model the result-ing weakly relativistic dilute pair population transforms theupward current region into a gyrotron which works due tothe comparably high energy (temperature) anisotropy of thepair population. The expectation in this case is that part ofthe auroral kilometric radiation is emitted in gyroradiation atsome harmonics. In this respect one may note that harmonicradiation has indeed been observed though not analyzed indetail. More important is, however, that temporarily highlyvariable structures have regularly been detected in the highresolution observations of FAST (Pottelette et al., 1999; Pot-telette & Treumann, 2005; Treumann et al., 2011) both in theupward and downward current regions.The conditions for sole gyrotron emission still seem to besevere. They become relaxed for moderately high harmonics.On the other hand, the gyrotron Dirac distribution may not beideally chosen to describe the real situation. It should thus betaken just as a first idealized step to a physical interpreta-tion of the fine structures which adds to some former models(Louarn et al., 1996; Treumann et al., 2011) from which theysubstantially di ff ffi cient when accounting forthe relativistic e ff ect. The same applies to the gyrotron emis-sion proposed here. So a combination of both is quite promis-ing. This will be deferred to a separate investigation. In thenext subsection we describe some ideas concerning the finestructure of AKR in view of gyrotron radiation. For data sets on AKR fine structure the reader may consultour above cited previous publications on this matter. We haveincluded some of them here in Figures 1 and 2. AKR is byno means a structureless banded emission close to the localgyrofrequency as suggested by any of the emission models.High temporal and frequency resolution of its upward currentregion (for the full upward current data set see Treumann etal., 2011, their Fig. 3, not included here) shows that the radi-ation consists of at least two components: a relatively broad-band rather weak and quasi-stationary emission spectrum onwhich a large number of intense drifting narrow-band emis-sions is superimposed as shown in the top of Fig. 1. Thesemay drift up and down in frequency at various frequencydrifts. In many cases high temporal resolution (Fig. 1, bot-tom) shows these little structures to move up and down infrequency and even to turn around or vanish at a certain placein the spectrum, often with the most intense radiation emit-ted just in the turn-around. We have tentatively in previouswork attributed this kind of motion to the presence of elec-tron holes which are known to drift up and down along themagnetic field. However, in view of the above problems withthe hole emission model we attempt to address the gyrotronmodel to these structures.Though it is by no means clear that the background com-ponent is indeed homogeneous – the highest resolution caseavailable to us seems to indicate that it is simply the unre-solved overlap of many drifting fine structures emitting ra-diation at larger distance from the spacecraft (FAST). Butconsider just the most intense banded radiation (Figs. 1 andFig. 2 in particular the t ∼ ∆ ω/ π ∆ t > / . =
100 kHz / s. The total emission band is restricted in this case tothe interval 425 < ω/ π <
440 kHz. In the topside auroral re-gion the curvature of the magnetic field is weak over a changein cyclotron frequency of this magnitude. The emission bandconsists of ∆ t (cid:46) . V A ∼ / s, the paired electrons have moved not much more than a distanceof ∆ s ≈
100 km along the magnetic field together with theAlfvén wave. This means that they have been very close totheir mirror point.One may speculate that the turn-around in the emissionwhere the intensity of radiation maximizes in a narrow emis-sion band of not more than ∆ ω/ π ∼ − ff erent magnetic flux tubes.Figure 2 is a particularly interesting case. No high res-olution data were available during this period. Here, thesteep nearly unresolved positive drifts are accompanied byweak emissions while the turn-arounds and substantially flat-ter negative drifts show intense emission when the electronsmove at much slower speed when slowly picking up thebounce speed. Of course, in this interpretation no directiv-ity of the emission is included as this has not been measured.In any case, the most intense emission is in the turn-around, a narrow band of bandwidth typically less than 1kHz. Let us assume that the emission is at the fundamental n = ff erent groups of electrons, spaced in frequency attheir turn-arounds have sightly di ff erent mirror points andthus probably have slightly di ff erent initial pitch-angles α orare in resonance with a di ff erent kAW. The latter is probableif the slopes of the emission bands at turn-around are di ff er-ent and they occupy the same flux tube. In the second halfof Fig. 1 the turn-arounds are parallel but at slightly di ff erentfrequencies separated by ∼ B ( R ) ≈ B / B ≈ .
44. This in the dipolefield implies an altitude of roughly h n = ≈ ∼ ∆ h (cid:46)
15 km, where inall these estimates we neglected the latitude dependence ofthe magnetic field.Each of the V-shaped emissions in such a model thencorresponds to a separate group of paired electrons movingalong in one flux tube with their resonant kAW. These groupsof pairs follow about regularly for a while every δ t (cid:38) . λ (cid:38) | n | =
1. When this is suppressed say either by the escape condi-tion with frequency mismatch ∆ < | n | = B ≈ h n = ≈ Sporadic very intense emission of electromagnetic waves, inparticular in the radio band, as is frequently observed in so-lar astrophysics and sometimes also from remote astronomi-cal objects, gives a clue to the understanding of the internalphysics of the emitting regions. It thus is useful as a con-venient remote probe. In basic electrodynamics such radia-tion relies on the simple gyro-synchrotron mechanism whichprincipally, because it is of higher order, is weak and, inorder to become intense, requires very large systems to in-crease the emission measure. Such systems evolve usuallyvery slowly such that sporadic emissions can hardly come upfor any short term intense variations like, for instance, therecently discovered and now quite frequently observed ex-tremely short broadband radio bursts.Moreover, at low energies particle scattering in the sourcesdo not contribute; they are spared for the much higher en-ergy range of X rays. Therefore emissions are sought forwhich are capable of causing intense fast sporadic radiationin the radio regime di ff erent from synchrotron emission, thefavoured mechanism in astrophysics since it is so simple.The electron cyclotron maser instability is probably the mostpromising in magnetized media. However, it requires partic-ular conditions set in the source which much be satisfied. Themost important is that the radiating plasma does not reabsorbthe emission, i.e. its absorption coe ffi cient must become ei-ther very small or negative. The latter case is realized in theECMI when the emitting electron population is lifted intoan excited state and at the same time there not much back-ground plasma is available to reabsorb the radiation. The lat-ter is most easily realized at higher harmonics since radiationat the fundamental is mostly damped due to trapping, quasi-linear quenching, and reabsorption.In the present letter we have attacked the problem of gen-eration of the pronounced fine structure superimposed on au-roral kilometric radiation (AKR) from the terrestrial topsideionosphere (or near-Earth magnetosphere) under disturbedauroral conditions. Generation of such fine structures in theemission is a non-trivial problem considering that the emis- sion band is just a few kHz wide, which is at most 1% of thebandwidth of AKR. It is believed that the latter is caused bythe general weakly relativistic pitch-angle distribution of au-roral electrons, a theory first proposed by Wu & Lee (1979)which accounts for the relativistic deformation of the reso-nance between electrons and free space modes through theinclusion of the dependence of the resonance on the trans-verse electron velocity β ⊥ appearing in the relativistic γ fac-tor. Inversion of the absorption property of the rather diluteplasma in this case is caused at the expense of the perpen-dicular (gyrational) energy of the plasma while still requir-ing the loss-cone as the agent of providing the demand offree energy stored in the inverted occupation of higher en-ergy levels. Since it is quite di ffi cult to believe that the mostintense and extremely narrow-band AKR emission is excitedby the global loss cone distribution, we have attempted an-other mechanism. Such a mechanism can possibly be foundin the generation of an attracting (negative) electrostatic po-tential Φ in the wake of moving electrons as proposed longago (Neufeld & Ritchie, 1955; Nambu & Akama, 1985) andrecently found (Treumann & Baumjohann, 2019) to be ap-plicable to magnetic mirror geometries in space plasma. Theattractive potentiantial is a single particle e ff ect which locallymodifies the plasma dielectric just outside the Debye sphereof the particle in cooperation (resonance) with a plasma wavemoving along the magnetic field and parallel to the electron.It requires that the phase velocity and the parallel speed of theelectron match, ω k / k ≈ v (cid:107) . Since the phase velocity is low, forbouncing electrons this condition implies that the e ff ffi ciently intense, is capable of simultaneously generatinga large number of harmonics it could possibly be responsiblefor their emission in not too strong magnetic field configura-tions such that the harmonic emission bands are not separatedtoo far in frequency. Acknowledgments
We acknowledge valuable discussions with the late JohannesGeiss, R. Nakamura, Y. Narita, and P. Zarka. RT acknowl-edges the contributions of R. Pottelette to all earlier work onthe subject of auroral kilometric radiation.
References
Ashour-Abdalla, M., and Thorne, R. M., The importance of elec-trostatic ion-caclotron instability for quiet-time proton auro-ral precipitation, Geophysical Research Letters 4, 45-48, 1977,https: // doi.org / / GL004i001p00045.Baumjohann, W., and Treumann, R. A., Basic Space PlasmaPhysics, Imperial College Press London, and World ScientificPress, Singapore, 1996; revised and enlarged edition, 2010.Bardeen J., Cooper, L. N., and Schrie ff er, J. R., Theoryof superconductivity, Physical Review 108, 1175-1204,1957,https: // doi.org / / PhysRev.108.1175.Bingham, R., Speirs, D., Kellett, B., Vorgul, I., Mcconville, S.,Cairns, R., Cross, A., Phelps, A., and Ronald, K. (2013). Labo-ratory astrophysics: Investigation of planetary and astrophysicalmaser emission. Space Science Reviews 178, 695-713.Cairns, R., Speirs, D., Ronald, K., Vorgul, I., Kellett, B., Phelps, A.,and Bingham, R. (2005). A cyclotron maser instability with ap-plication to space and laboratory plasmas. Physica Scripta 2005,23.Carlson, C. W., Pfa ff , R. F., and Watzin, J. G., The Fast Auro-ral SnapshoT (FAST) mission, Geophysical Research Letters 25,2013-2016, 1998, https: // doi.org / / // doi.org / / PhysRev.104.1189.Dory, R. A., Guest, G. E., and Harris, E. G., Unstable elec-trostatic plasma waves propagating perpendicular to a mag-netic field, Physical Review Letters 14, 131–133, 1965,https: // doi.org / / PhysRevLett.14.131.de Feraudy, H., Pedersen, B. M., Bahnsen, A., and Jes-persen, M., Viking observations of auroral kilometric radi-ation from the plasmasphere to night auroral oval sourceregions, Geophysical Research Letters 14, 511-514, 1987,https: // doi.org / / GL014i005p00511.Gillespie, K., Speirs, D., Ronald, K., Mcconville, S., Phelps, A.,Bingham, R., Cross, A., Robertson, C., Whyte, C., and He, W.(2008). 3D PiC code simulations for a laboratory experimentalinvestigation of auroral kilometric radiation mechanisms. PlasmaPhysics and Controlled Fusion 50, 124038.Einstein, A., Zur Quantentheorie der Strahlung, Mitteilungen derPhysikalischen Gesellschaft Zürich 18, 47-62, 1916.Ergun, R. E., Carlson, C. W., McFadden, J. P., Mozer, F. S.,Muschietti, L., Roth, I., and Strangeway, R. J., Debye-ScalePlasma Structures Associated with Magnetic-Field-Aligned Electric Fields, Physical Review Letters 81, 826-829, 1998,https: // doi.org / / PhysRevLett.81.826.Ergun, R. E.; Carlson, C. W.; McFadden, J. P.; Mozer, F. S.; De-lory, G. T.; Peria, W.; Chaston, C. C.; Temerin, M.; Roth, I.;Muschietti, L.; Elphic, R.; Strangeway, R.; Pfa ff , R.; Cattell,C. A.; Klumpar, D.; Shelley, E.; Peterson, W.; Moebius, E.;Kistler, L., FAST satellite observations of large-amplitude soli-tary structures, Geophysical Research Letters 25, 2041-2044,1998, https: // doi.org / / // doi.org / / BF01031607.Gurnett, D. A., The Earth as a radio source: Terrestrial kilometricradiation, Journal of Geophysical Research 79, 4227-4238, 1974,https: // doi.org / / JA079i028p04227.Fröhlich, H., Interaction of electrons with lattice vibrations, Pro-ceedings of the Royal Society of London, Series A, 215, 291-298, 1952, https: // doi.org / / rspa.1952.0212.Kennel, C. F., and Petschek, H. E., Limit on stably trapped par-ticle fluxes, Journal of Geophysical Reseaarch 71, 1-28, 1966,https: // doi.org / / JZ071i001p00001Krall, N. A., and Trivelpiece, A. W., Principles of Plasma Physics,McGraw-Hill, New York, 1973.LaBelle, J., and Treumann, R. A., Auroral Radio Emissions, 1.Hisses, Roars, and Bursts. Space Science Reviews 101, 295-440,2002, https: // doi.org / / A:1020850022070.Louarn, P., and Le Quéau, D., Generation of the Auroral Kilo-metric Radiation in plasma cavities - I. Experimental study, II.The cyclotron maser instability in small size sources, Planetaryand Space Science 44, 199-224, https: // doi.org / / // doi.org / / GL014i004p00443.Lundin, R., Eliasson, L., Haerendel, G., Boehm, M., and Hol-back, B., Large-scale auroral plasma density cavities observedby Freja, Geophysical Research Letters 21, 1903-1906, 1994,https: // doi.org / / // doi.org / / // doi.org / / S1743921309029470.Nambu, M. and Akama, H., Attractive potential between reso-nant electrons, Physics of Fluids 28, 2300-2301, 1985, https: // doi.org / / Narita, Y., O. W. Roberts, Z. Vörös, and M. Hoshino, Transportratios of the kinetic Alfvén mode in space plasmas, Frontiers inPhysics 8, 00166, 2020, https: // doi.org / / fphy.2020.00166.Narita, Y., Baumjohann, W., and Treumann, R. A., Pairing in kineticAlfvén waves, in preparation, 2020.Neufeld, J., and Ritchie, R. H., Passage of charged particlesthrough plasma, Physical Review 98, 1632-1642, 1955, https: // doi.org / / PhysRev.98.1632.Paschmann, G., Haaland S., and Treumann, R. A. (eds.), Auro-ral Plasma Physics, Space Sciences Series of ISSI Volume 15,Springer-Verlag, New York, 2003.Pottelette, R.; Ergun, R. E., Treumann, R. A., Berthomier, M.,Carlson, C. W., McFadden, J. P., and Roth, I., Modulatedelectron-acoustic waves in auroral density cavities: FAST ob-servations, Geophysical Research Letters 26, 2629-2632, 1999,https: // doi.org / / // doi.org / / // doi.org / / PhysRevLett.2.504.Speirs, D., Bingham, R., Cairns, R., Vorgul, I., Kellett, B., Phelps,A., and Ronald, K. (2014). Backward wave cyclotron-maseremission in the auroral magnetosphere. Physical review letters113, 155002.Speirs, D., Mcconville, S., Gillespie, K., Ronald, K., Phelps, A.,Cross, A., Bingham, R., Robertson, C., Whyte, C., and Vorgul,I. (2008). Numerical simulation of auroral cyclotron maser pro-cesses. Plasma Physics and Controlled Fusion 50, 074011.Speirs, D., Vorgul, I., Ronald, K., Bingham, R., Cairns, R., Phelps,A., Kellett, B., Cross, A., Whyte, C., and Robertson, C. (2005). Alaboratory experiment to investigate auroral kilometric radiationemission mechanisms. Journal of plasma physics 71, 665-674.Sprangle, P., and Drobot, A. T., The linear and self-consistent non-linear theory of the electron cyclotron maser instability, IEEETransactions on Microwave Theory and Techniques MTT-25,528-544, 1977, https: // doi.org / / TMTT.1977.1129151.Sprangle, P., Granatstein, V. L., and Drobot, A. T., The electroncyclotron maser instability, Journal de Physique 38, Colloque C-6, Supplement 12, C6-135, 1977.Treumann, R. A., The electron-cyclotron maser for astrophysicalapplication, Astronomy and Astrophysics Reviews 13, 229-315,2006, https: // doi.org / / s00159-006-0001-y.Treumann, R. A., Baumjohann, W., and Pottelette, R., Electron-cyclotron maser radiation from electron holes: upward cur-rent region, Annales Geophysicae 29, 1885-1904, 2011,https: // doi.org / / angeo-29-1885-2011; Electron-cyclotron maser radiation from electron holes: downwardcurrent region, Annales Geophysicae 30, 119-130, 2012,https: // doi.org / / angeo-30-119-2012.Treumann, R. A., and Baumjohann, W., Plasma wave medi-ated attractive potentials: a prerequisite for electron com-pound formation, Annales Geophysicae 32, 975-989, 2014,https: // doi.org / / angeo-32-975-2014.Treumann, R. A., and Baumjohann, W., Anisotropic Jüttner (rela-tivistic Boltzmann) distribution, Annales Geophysicae 34, 737-738, 2016, https: // doi.org / / angeo-34-737-2016.Treumann, R. A., and Baumjohann, W., Electron pairing in mirror modes: surpassing the quasi-linear limit, Annales Geophysicae37, 971-988, 2019, https: // doi.org / / angeo-37-971-2019.Twiss, R. Q., Radiation transfer and the possibility of negative ab-sorption in radio astronomy, Australian Journal of Physics 11,564-579, 1958, https: // doi.org / / PH580564.Wu, C. S., and Lee, L. C., A theory of the terrestrial kilo-metric radiation, Astrophysical Journal 230, 621-626, 1979,https: // doi.org / /157120.