MManuscript prepared for Journalnamewith version 3.1 of the L A TEX class copernicus2.cls.Date: 3 September 2020
Topside Reconnection
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 .– It is proposed that reconnection would be a mainmechanism governing the plasma processes on auroral timescales in the topside ionosphere / high-latitude magnetospheretransition. It occurs in the downward current region betweentwo narrow parallel closely spaced though separated down-ward current sheets. The field-aligned currents are carried byupward cold upper-ionospheric electrons closing the upwardcurrent in an adjacent region. This local process does primar-ily not a ff ect the ambient field but generates an anomalousdi ff usivity.In a recent paper (Treumann & Baumjohann, 2017) wesuggested that strong-guide field reconnection may play arole in the generation of radiation in the topside auroral iono-sphere by the electron cyclotron maser instability mechanism(ECMI) (Wu & Lee, 1979; Melrose, 1989; Treumann, 2006).This idea was based on the assumption that at the boundarybetween the upward and downward current regions the mag-netic fields of kinetic Alfvén waves might undergo reconnec-tion, causing electron exhausts which are similar to electronholes while being of larger scale. Such a mechanism seems tobe appropriate to explain intense emission in the auroral kilo-metric radiation band (Gurnett, 1974). In a subsequent paperwe explored a particular model of stronger amplification ofthe radiation if electron pairing would occur in the vicinityof the electron mirror points along the strong auroral mag-netic field. This brings up the question whether, independentof the generation of radiation, reconnection might not be thedominant process of plasma dynamics in the topside auroralregion.The topside auroral region is characterized by a numberof properties which at first glance do not seem in favourof reconnection. In order to have reconnection one needscontact between anti-parallel magnetic fields and plasma in-flow perpendicular to the field, as is inherent to all the ba-sic reconnection models (see Vasyliunas, 1975; Treumann& Baumjohann, 2013, for reviews separated by 40 years).The auroral region, at the contrary, hosts a very strong mag-netic field which on the scales of aurora is parallel (exceptfor some weak inclination and systematic geographic vari- ation). It does not change sign across the auroral region onone hemisphere. Thus it seems highly improbable that such afield would undergo any reconnection and rearrange in someviolent manner. Indeed, it does not. In order to rearrangeit requires very strong external basically mechanical forcesto twist or wrap it around. Such forces would be related toextremely strong currents which in the topside ionosphere-magnetosphere transition renders them completely improba-ble. They require conditions which are presumably realizedin the lower solar atmosphere where the solar photosphericconvection network rotates the frozen-in magnetic field atfrequency of few minutes around causing the field to be-come highly warped into a spiral which stretches out intothe corona and solar wind. The strong geomagnetic field intopside auroral region is in contrast fixed to the inert iono-sphere and body of the earth. Deforming it substantially re-quires very strong outer forces which happens very rarely. Itmay, however, play a decisive role as catalyst of reconnectioncaused by other means.The auroral region is comparably narrow in (geo-graphic / geomagnetic) latitude while somewhat broader in itslongitudinal extension. In each auroral event, it divides intotwo sections, one carrying downward fluxes of medium en-ergy (cid:15) e ∼
10 keV electrons corresponding to upward field-aligned currents, the other hosting upward electron fluxesof energies (cid:15) e (cid:46) downward field-aligned currents. There is a spectrum of magnetic variations(LaBelle & Treumann, 2002) partially related to these cur-rents which is interpreted di ff erently in terms of waves be-low the local electron cyclotron frequency ω < ω c which atthe lower spectral end has substantial amplitudes δ B usuallyinterpreted as caused by fluctuations of the main field B im-posed by the magnetosphere. Still, the amplitudes are smallin the sense that δ B (cid:28) B . The former section has severaltimes larger spatial extension than the latter. The two regionsoccur always in tandem, typical for a closed current-returncurrent system and, in most cases, not as a single current pairbut in groups of several up-down pairs, generally being sep-arated by a region of no auroral current activity. The phe- a r X i v : . [ phy s i c s . s p ace - ph ] S e p R. A. Treumann & W. Baumjohann: Topside Reconnectionnomenology and models have been described in Paschmannet al. (2003, chpts.1-8).There is no obvious local reason for field-aligned currentsin the topside auroral ionosphere to be dispersed in the man-ner observed. Their most reasonable driving source is recon-nection in the tail current sheet, however, which is well estab-lished. Upward currents / downward electrons originate fromelectron acceleration in the central tail plasma sheet. Theyflow down along the newly reconnected closed magnetic fieldinto auroral latitudes causing the upward field aligned cur-rents. Upward low energy electron fluxes belong to the down-ward closure currents and result from moderately acceleratedionospheric electrons present here in su ffi ciently large num-bers. How this acceleration happens in detail remains unclearbut can be taken as an observational fact. More than one up-down pair of currents indicates multiple tail reconnection. Itseems that this is the only causally satisfactory picture. (Itsgross geometry is depicted in Fig. 1.)Some of the most violent auroral processes result from thedynamics of the downward / upward electron fluxes and therelated upward / downward field-aligned currents. (A full se-quence of FAST observations when crossing a topside ac-tive auroral region during a substorm is given in Fig. 2.)Of course, since reconnection in the tail is non-stationary,its longer temporal scale folds into the internal processescaused by the fluxes and currents. It modifies those whilecan be considered as waves flowing along the field with non-stationary currents coming in field-aligned electromagneticwave pulses. These are assumed as belonging to one of the(kinetic) Alfvén modes. Thus, on the time-scale of the lat-ter, auroral dynamics will be related to the electromagneticstability of the field-aligned current pulses. The observations concerning auroral electron fluxes and therelated currents are the following:
Downward electrons / upward currents occupy an extendedspatial interval of low density and barely structured fluxes.The variation of the perpendicular (to the main field) mag-netic field is smooth; it changes about linearly from − δ B ⊥ to + δ B ⊥ signalling that the spacecraft has crossed a homoge-neous broad structureless upward sheet current carried by theas well structureless medium energy electron flux which, inthe energy-time spectrum occupies a narrow band of constantenergy and small energy spread.Absence of an ionospheric electron background at (FAST)spacecraft altitude either suggests that the ionosphere doesnot reach up to those altitudes ( ∼ − ff usive model of the ionosphere, is inter-preted as presence of a field aligned electric potential which holds the ionospheric electrons down while attracting mag-netospheric electrons. The validity of such an assumptioncan be questioned in terms of tail reconnection as the inflowof reconnection accelerated electrons from the tail does notrequire such an electric potential field, the origin of whichis di ffi cult to justify over a region of upward current exten-sion while being natural when considering tail reconnectionwhere it simply maps the large reconnection a ff ected inter-val of the cross tail current down into the ionosphere. Thesmall number of downward electrons does not require anypresence of electric fields. The flux consists of nearly mono-energetic auroral electrons. These form a field-aligned beamand are accompanied by observed low frequency Langmuir-wave excitation which allows for the determination of thebeam density being roughly N ↓ ≈ m − (one electron percubic centimetre). Figure 3 shows simultaneous upward / downward FAST mea-surements of electron fluxes when crossing a very activesubstorm topside auroral ionosphere. The upward-electrondownward-current region behaves di ff erently. Its spatial ex-tension is narrow. In view of the electron flux it consistsof a large number of very closely spaced spikes. The fluxin each spike (generally) maximizes at the lowest energies (cid:15) e (cid:46) . N ↑ ∼ m − (ten per cubic centimetre) orhigher. The total integrated up and down currents must besimilar for perfect closure. This is however not guaranteedfor the divergence of currents in the ionosphere perpendicu-lar to the magnetic field, current dissipation, and the high spa-tial structuring of the downward currents such that it cannotbe checked whether the indication of the di ff erent downwardcurrent sheets all belong to closure of the single upward cur-rent. Some uncertainty in comparison remains, which how-ever for our purposes does not matter.The important observation is the high local structure ofthe downward currents, their obvious spatial closeness, andtheir di ff erences in energies and flux level which is reflectedin both the flux fluctuations across the narrow downwardcurrent region, and in the high spatial fluctuation of themain-field-perpendicular magnetic component b ⊥ (from hereon denoting the magnetic variation δ B ⊥ = b ⊥ ) which indi-cates the crossing of many downward current sheets or fil-aments. All these downward currents flow parallel while be-ing closely spaced in the direction perpendicular to the mainfield B Figure 1.
Schematic of connection between tail reconnection and auroral topside current system for one single tail X-point (after Treumannet al., 2011a). Downward and upward field-aligned electron fluxes are indicated in the topside ionosphere. They correspond to upward anddownward field-aligned currents. Naturally, due to the geometry of the tailward source, the downward fluxes are distributed over a largespatial domain, while the returning upward fluxes occupy a narrow latitudinal interval only on the northern edge. because they transport current. Lorentz attraction forces thecurrents to approach each other to form a broad unstructureddownward current sheet. This is, however, inhibited by thestrong main auroral geomagnetic field B . The argument thatthis should also happen in the upward current region fails be-cause the current sheet there is broad by its origin from thetail reconnection site.Since anti-parallel currents reject each other the transitionregion between upward and downward currents is quiet. Thisis seen in panels ( b − f ) of Fig. 2 and is in contrast to ourprevious investigation where we assumed that reconnectionwould happen there between parallel kinetic Alfvén waves.The Lorentz force between two equally strong sheet currents J (cid:107) is J (cid:107) × b ⊥ = −∇ ⊥ b ⊥ /µ (1)where b ⊥ is the magnetic field between the two currents (inthe following we suppress the index ⊥ on the magnetic fieldcomponent b ), and ∇ ⊥ refers to the gradient in the direc-tion from current sheet to current sheet. The current consists,however, of gyrating electrons whose Lorentz force is thecross product of the azimuthal gyration speed times the verystrong stationary field with gradient ∇ c taken only over thegyro-radius r ce = v e ⊥ /ω ce of the electrons. For a separation ofthe sheet current exceeding the electron gyroradius and lowcurrent density the sheet currents will approach each otheronly on very long di ff usive time scales of no interest. For a thin current sheet only a few gyroradii thick the condition forthis time to be long is simply that the electron inertial lengthexceeds the gyroradius or v e ⊥ / c (cid:28) ω ce /ω e (2)which holds under very weak conditions in the topside au-roral ionosphere. This implies that downward current sheetsseparated by say an electron inertial length λ e = c /ω e will notmerge under no circumstance. They remain separated overthe observational spacecraft crossing time scales. It is theirsecondary magnetic field b which will undergo reconnec-tion without a ff Figure 2.
Full sequence of FAST measurements across dow-up-auroral current system on 02-01-1997 (after Treumann et al., 2011b). ( a )Magnetic field component b ⊥ transverse to main field B , ( b ) electric field fluctuation wave form δ E , ( c ) low frequency electric fluctuationspectrum, ( d ) high frequency electric spectrum, showing emission of auroral kilometric radiation bands ( e ) electron energy spectrum, ( f )electron flux versus pitch-angle, ( g ) ion energy flux, ( h ) ion flux versus pitch-angle. The most intriguing part here is the smoothness of themagnetic signature of the upward current in its linear course showing that the upward current is a broad homogeneous current sheet. Thedownward current region (DCR) flanks the upward current region to both its sides, is comparably narrow in its spatial extent, and exhibitsstrong current and flux variations. This is seen in the electron flux panels ( e − f ). Downward fluxes around few keV are relatively smoothindicating a relatively stable tail reconnection over observation time, upward fluxes have maximum at low energy and are highly variable intime and space.The magnetic field being the integrated response to the spatial flux fluctuations exhibits a much smoother course which isinverse with respect to that of the upward current thus indicating the reversed current direction. Note the low energies of the upward electronfluxes in panel five as well as the clear separation of upward and downward fluxes as seen in the left part. In the complementary wave picture of field-aligned cur-rents in the auroral region, the current is carried by (kinetic)Alfvén waves in the frequency range well below the localion-cyclotron frequency. In addition, a large number of low-frequency electromagnetic waves are known to be presentthere (LaBelle & Treumann, 2002). We are in the down-ward current region with highly sheared upward particle flowalong the magnetic field consisting of moderately fast elec-trons and much slower ions. Such flows are capable of gener-ating Alfvén waves (Hasegawa & Uberoi, 1982) on perpen-dicular scales of the ion inertial length λ i = c /ω i = λ e √ m i / m e and below and long wavelength parallel to the ambient field.For the current-carrying electrons such waves are about sta-tionary magnetic structures.These Alfvén waves cannot be body waves like in the solarwind (Goldstein et al., 2005; Narita et al., 2020) because theyare strictly limited to the narrow field-aligned current sheets.Since they propagate in the strong auroral geomagnetic field,they are rather di ff erent from the usual kind of kinetic Alfvénwaves which one refers to in solar wind turbulence (Gold-stein et al., 2005), where the magnetic field ist very weakand the turbulence is dominated by the mechanics of the flow(Maiorano et al., 2020). There the ion-temperature plays animportant role imposing kinetic e ff ects on the wave.Under auroral conditions, in particular close to the iono-sphere, the magnetic field is so strong that thermal ion e ff ectson the wave are barely important. Their mass e ff ect enters theAlfvén speed. Instead, however, under those conditions elec-tron inertia on scales λ i ∼ ∆ (cid:38) λ e = c /ω e below the ion scalecomes into play. For su ffi ciently narrow field-aligned currentsheets of width the order of inertial scales, the field does notallow the electrons to leave their flux tube unless they havelarge perpendicular moment. Field-aligned electrons remaininside their gyration flux tube, and the currents cannot re-act to merge with neighbouring parallel current sheets. TheLorentz force on the field-aligned current in the magneticfield of its neighbour is not strong enough to move the cur-rents. In this case the kinetic Alfvén waves transporting thecurrents in pulses become inertially dominated with disper-sion relation ω = k (cid:107) V A (cid:16) + k ⊥ ρ i (cid:17) + k ⊥ λ e (3)where ρ i is some modified ion gyro-radius (cf., e.g. Baumjo-hann & Treumann, 1996) containing kinetic temperaturecontributions. For the cold ions in the topside auroral iono-sphere the term containing ρ i in the numerator vanishes. Thekinetic Alfvén wave under those conditions becomes an in-ertial or shear wave. It propagates at a reduced though still fast speed along the magnetic field, being of very long par-allel wavelength. It also propagates slowly perpendicular tothe magnetic field at short wavelength λ ⊥ ∼ λ e / π . It is, inprinciple, this wave which carries the current. Thus the cur-rent is not stationary on time scales long compared to theinverse frequency ∆ t > ω − but can be considered stationaryfor shorter time scales ω ∆ t < ∂ω∂ k (cid:107) = V A (cid:113) + k ⊥ λ e , ∂ω∂ k ⊥ = − ∂ω∂ k (cid:107) k (cid:107) k ⊥ (cid:104) + / ( k ⊥ λ e ) (cid:105) (4)Energy transport in the perpendicular direction is smallerthan parallel by the ratio of wave numbers.Repeating that we are in the downward current upwardelectron flux region causality requires that the upward elec-trons carry information from the ionosphere to the magne-tosphere. Hence the kinetic Alfvén waves in this region alsopropagate upward being produced in the topside by the trans-verse shear on ion-inertial scales below λ i . Assuming stationarity of the field aligned current J = J (cid:107) B / B in two adjacent but separated parallel sheets andassuming, for simplicity, that the two currents are of equalstrength, reconnection will occur in the central region of sep-aration of the current sheets. (Figure 4 shows a two-crosssection schematic of the downward current-field configura-tion for two closely spaced current sheets.) Here the twomagnetic fields of the field-aligned currents are antiparallel.According to the above discussion, we are in the downwardcurrent region (as in the case of our previous radiation modelTreumann et al., 2011b). In fact an analogue model wouldapply to the upward current region. The current flows in di-rection z ; the direction of y is longitudinal (eastward), x islatitudinal (northward). If the sheet ist extended mostly in y the antiparallel magnetic fields are in y along the sheets. Theywill touch each other and reconnect between the two sheetsthereby forming reconnection X-points with field component ± b x and extended magnetic field free electron exhausts in x and in y , which contain the local main-field parallel recon-nection electric field, and accelerate electrons along the am-bient field. Tjhese exhausts will propagate along the mainfield together with the wave. Plasmoids might also form inthe separation between the sheets perpendicular to the am-bient field, and the presence of the strong ambient field willimpose electron gyration and scatter of electrons causing sec-ondary e ff Figure 3.
Sequence of downward (top) and upward (bottom) auroral electron fluxes observed by FAST on 02-07-1997 in the topside auroralregion when crossing a substorm aurora (after Treumann et al., 2011a). The sequence distinguishes nicely between the intense downwardelectron fluxes at energies (cid:15) e ∼
10 keV and downward fluxes at energies (cid:15) e < . There are two essential di ff ± b .This current builds up dynamically and locally duringthe reconnection process itself when the two kinetic Alfvénwaves slowly move perpendicular. This is a di ffi cult dynami-cal problem in that reconnection will set on when the encoun-tering magnetic fields exceed some threshold. Since electronsin this region are magnetized by the strong ambient field,they gyrate but do not take notice of the weak field b of thekinetic Alfvén waves which is transported across the sepa-rating region by the perpendicular phase and group speeds ofthe waves to get into contact and merge.The reconnection process is thus solely between the twowaves, primarily not a ff ecting the ambient field and not basedon any real central primary current sheet. Observations so fardo not resolve the magnetic nor the particle e ff ects of suchfictitious return currents though some of the structure seen inthe low energy electron fluxes in Fig. 3 could be interpretedas such without proof. In fact, in order to avoid formation ofthe fictitious return current, which would imply that this cur-rent would be equally strong in the gap between the currentsheets, reconnection is required over the full length of half awavelength along z . Thus it necessarily generates elongatedfield-aligned vertical X-lines and electron exhausts in z . All these e ff ects are of vital interest. However, one particu-larly interesting question concerns the dissipation producedby this kind of reconnection. It is frequently argued that itleads to sliding of main-field field lines. In order to under-stand such a mechanism one needs to know the anomalousresistance caused by reconnection. In electrodynamic for-mulation, reconnection is conventionally dealing solely withthe merging and energy transfer of fields. The microscopicmechanism of energy transfer is accounted for in the trans-port coe ffi cients. Hence the appropriate way of inferring theirvalue is referring to the electromagnetic energy exchange.This leads to the application of Poynting’s theorem ∂ b ∂ t = − µ η an J (cid:107) − ∇ ⊥ · (cid:16) E (cid:107) × b (cid:17) − ∇ ⊥ · (cid:16) E rec × B (cid:17) (5)where the contribution of the electric field to the left-handside is neglected as it is relativistically small, and b is themagnetic field of the field-aligned current. It allows for a convenient estimate of the anomalous resistivity η an in re-connection without going into any microscopic detail of themechanism of its generation. The electric field in this expres-sion is along the ambient magnetic field, essentially beingthe electric field of the kinetic Alfvén wave. Estimates of thisparallel field have been provided by Lysak & Lotko (1996)and were taken as the important agent for accelerating auro-ral electrons.The above expression shows that reconnection in this caseis a two-step process. In the first step the parallel field E (cid:107) along the ambient magnetic guide field sets up reconnection.In the second step the reconnection electric field E rec andexhaust have evolved. The cross-product with the main mag-netic field then modifies the dynamics of the exhaust. In this subsection we are not interested in this e ff ect here as itis overwritten once reconnection really sets on but enters inthe determination of the perpendicular inflow speed. It causesit to be di ff erent from tailward reconnection. Instead we pro-ceed to an estimate of the anomalous collision frequency.The parallel electric field E (cid:107) of the kinetic Alfvén waveplays an important role in the first step of the topside re-connection process. Since this field is parallel to the ambientgeomagnetic field B , the cross product with the wave mag-netic field is responsible for the two current-sheet magneticfield components ± b to approach each other in the regionbetween the sheets. Hence, referring to this fact, the secondterm on the right can be expressed through the perpendicularvelocity V ⊥ = E (cid:107) × b / b , and we have ∇ ⊥ · (cid:16) E (cid:107) × b (cid:17) = ∇ ⊥ · (cid:16) V ⊥ b (cid:17) (6)In order to get some information about the perpendicu-lar velocity V ⊥ which according to our coordinate systempoints to the centre of the region which separates the two cur-rent sheets, i.e. along y , we refer to the wave picture, notingthat these pictures are equivalent: the field-aligned current J (cid:107) is carried by (upward topside ionospheric) electrons, on theother hand these electrons are transported (or pushed) by thekinetic Alfvén wave. In fact, of course, V ⊥ is counted fromeach of the two parallel currents as pointing to the center ofthe separating sheet. It thus in our water-bag model changesabruptly sign in the center where due to the two antiparallelmagnetic fields which collide there a fictitious weak returncurrent of strength j (cid:107) ≈ b /µ δ arises, with δ the fictitiouswidth of this narrow current layer which we do not explicitlyconsider. The simplest is in our water-bag model to assumethat for closely separated parallel current sheets we have es-sentially δ → α ∆ , with α (cid:46)
1, and a return current distributedover almost the entire separation width. One should also keepin mind that any field-aligned current carried by the Alfvénwave is a current pulse with both E (cid:107) and b changing direc-tion (oscillating) over half the wavelength. Thus V ⊥ ± V ⊥ are just the per-pendicular phase speeds of the kinetic Alfvén waves on thetwo adjacent current sheets V ⊥ ∼ ω k ⊥ ≈ V A (cid:113) + k ⊥ λ e k (cid:107) k ⊥ (cid:28) V A (7)This velocity apparently diverges for k ⊥ → k ⊥ (cid:44) k ⊥ → k ⊥ λ e ∼ ∼ V A / √ ∼ − V A ( k (cid:107) λ e ) / / . However, since V ⊥ indeed transportsnot only the field but also energy, one may argue that the useof the latter expression would be more appropriate than thephase speed. Since this does not make any big di ff erence forour purposes, we in the following for reasons of simplicityunderstand V ⊥ as phase speed. For more precise expressionsone may replace it in the following with the perpendiculargroup speed.The velocity V ⊥ is small because k (cid:107) (cid:28) k ⊥ , i.e. the kineticAlfvén wave is long-wavelength parallel to the ambient fieldbut of short perpendicular wavelength, a very well-knownproperty. Moreover, V ⊥ ( z ) may vary along the ambient fieldbut, in the frame of the wave, which corresponds to a water-bag model, is constant in the perpendicular direction. Hence,of the above vector product just remains the variation of themagnetic field b ( x ) over the distance between the two currentsheets. This insight enables us to rewrite Poynting’s equationas ∂ b ∂ t ≈ − µ η an J (cid:107) − V ⊥ ∂ b ∂ x (8)which, assuming a stationary state, enables us to estimate theanomalous resistivity of stationary reconnection (in the waveframe) where the inflow of magnetic energy attributed bythe current, i.e. the field-aligned electron flux whose originis found in reconnection in the magnetotail, is balanced byanomalous energy transfer to the plasma in the region sep-arating the two current sheets. Putting the left-hand side tozero we thus find that in this kind of topside reconnection theanomalous resistivity is bound from above as η an (cid:46) V A (cid:113) + k ⊥ λ e k (cid:107) k ⊥ b µ ∆ J (cid:107) ≈ V A b µ ∆ J (cid:107) k (cid:107) λ e (1 + k ⊥ λ e ) / (9)where ∆ is the spatial separation of the two field-aligned cur-rent sheets, and we have taken into account that each of thetwo identical current layers contributes a field b . The secondpart of this expression makes use of the perpendicular groupspeed. This resistivity is small as k (cid:107) λ e / ∆ but finite. It givesrise to a finite di ff usion coe ffi cient that can be interpreted asan anomalous di ff usivity for the ambient magnetic field in the auroral topside ionosphere, caused by topside reconnectionbetween anti-parallel current sheets in the downward currentregion. We might note at this occasion that the restriction tothe downward current region is motivated by the observationof narrow current sheets in the downward current region. Ob-servations do not suggest that similarly narrow current sheetsevolve in the upward current region. If this would be the case,the same arguments would apply there, causing reconnectionand a similar anomalous resistivity.What concerns the spatial separation of the current sheets(see Fig. 3), the best available observations (FAST) do notresolve any single sheets; it can however be assumed thattheir scales are the order of or below the ion-inertial length,such that ∆ (cid:46) λ i ∼ several to many λ e . This may overesti-mate the real value but has been accounted for in writing theexpression as an upper limit. Determination of the anoma-lous resistivity thus requires knowledge of the field alignedcurrent density, current sheet separation, and the transversemagnetic field component of the sheet current. We then canestimate the anomalous collision frequency ν an = η an (cid:15) ω e inthis kind of reconnection ν an (cid:46) V A /α λ e (cid:113) + k ⊥ λ e k (cid:107) ∆ k ⊥ λ e (10)where we used that J (cid:107) ≈ b /α ∆ . Note that V A = B / √ µ m i N is based on the ambient magnetic field and plasma density.This simple estimate shows that reconnection in this casecan, under stationary condition be described as being equiv-alent to a di ff usive process based on the anomalous collisionfrequency which is provided by the merging of the transversemagnetic fields of the two neighbouring field-aligned currentsheets. Since the related di ff usivity is felt in the entire re-gion it is remarkable that it could e ff ect also the main am-bient guide field. In other words, topside reconnection couldbecome responsible for di ff usion of the main magnetic fieldlines in a locally restricted domain possibly causing e ff ectson a larger scale in the auroral region.Real reconnection will not occur between field-alignedcurrent sheets of same strength. Thus the above resistivityrespectively the collision frequency must be reduced by an-other factor proportional to the involved current and fieldfractions. So far we just investigated the energy balance in order to ob-tain an anomalous collision frequency in this kind of recon-nection. Reconnection however manifests itself in X pointsgenerating transverse magnetic fields and in addition electricfields. Since there is no primary return current flowing, it can-not be used as input into the two-dimensional reconnectionequation for the vector potential A z ∇ A z = − µ j z ( x ) , ∇ = ( ∂ x ,∂ y , , j z ( x ) = − (cid:15) b ( x ) /µ ∆ Figure 4.
Schematic of the field configuration between two parallel field aligned flux tubes in the downward current region.
Left : Geometryalong the ambient field. Currents are in red. Included would be the (red dashed) fictitious return current which locally would correspondto the antiparallel wave magnetic fields ± b . This current would be local over the wavelength of the inertial Alfvén wave. In any stationaryreconnecting current picture it would be this current whose magnetic field reconnects. However, here this current does not exist in the exhaust.It is completely reconnected and gives rise to the reconnection electric field E z instead (dashed red) in the exhaust along the main magneticfield. Electrons are directly accelerated by it along B . Right : Reconnection geometry with perpendicular velocity V ⊥ , field free exhaust,reconnection fields ± b x indicated, and E z . The anomalous collisions caused in the exhaust volume also permit for weak di ff usion of theambient field. This may cause what is believed to be magnetic field di ff usion, a very slow process compared to the wave / current inducedspontaneous reconnection. without prescribing the built-up of the central current profile j z ( x ), which is possible only when assuming that the b is in-dependent of x , in which case it provides the usual stationarytearing mode solution (see, e.g., Schindler, 1974) rewrittenfor electrons alone. Under these simplifying restrictions thetwo components of the reconnected magnetic field includingthe X point are given by b = ( ∂ y A z , − ∂ x A z , ffi ces for our qualitative considerations. The a priori as-sumption of a return current is, however, incorrect. On thetopside there may weak local return currents exist filling theseparations between the narrow downward current sheets, butthe main return current flows in the upward current regionand is distributed over a wide domain. Hence just a fraction (cid:15) of return current can flow in the gap, as included in the lastexpression. The electric field in this case primarily has onlyone component, which is along the main field and is givenby E z = − ∂ t A z −∇ U where U is the scalar electrostatic gaugepotential which may occur if an inhomogeneity exists or thesystem is not ideally symmetric. This field adds to the fieldaligned kinetic Alfvén wave electric field and contributes toelectron acceleration. It is the wanted reconnection electricfield and can be much larger than the small linear wave elec-tric field. Unfortunately its precise knowledge requires so-lution of the equation for the vector potential A z and someinterpretation of the time derivative operator. The latter canbe transformed into a spatial derivative ∂ t = ± V ⊥ · ∇ A z ( x , y ).The important conclusion in the case of topside reconnec-tion is rather di ff erent from usual reconnection. It tells thatthe exhaust is, over half the wavelength of the inertial Alfvénwave free of wave magnetic fields b , while being boundedby the reconnected wave fields ± b x . The exhaust instead con-tains the reconnection electric field, by being along the mainfield, does directly contribute to acceleration respectively de-celeration of electrons (and also ions) along the main mag-netic field, one of the most important and still unresolvedproblems in auroral physics. There acceleration is attributedto a variety of waves, reaching from kinetic Alfvén throughwhistlers and several electrostatic waves to electron and ionholes. Except for the latter nonlinear structures, all waveelectric fields are quite weak, and in addition fluctuate. Ac-celeration thus becomes a second order process.In case of the topside reconnection, a mesoscale first-orderelectric field E z is produced which directly accelerates parti-cles, depending on its direction along the main field. More-over, the source of the accelerated particles is the gap regionbetween the two current sheets, the so-called exhaust, suchthat the kinetic Alfvén wave electric field and the reconnec-tion electric field do barely interfere. Hence the full strengthof the reconnection exhaust field acts accelerating. One maythus conclude that topside reconnection, if it takes place, willsubstantially contribute to auroral particle acceleration.In order to circumvent the above named di ffi culty of calcu-lating A z and to obtain an estimate of the reconnection elec-tric field, we may return to the induction equation in its inte-gral form where the electric field is given by the integral overthe surface of the reconnection site (cid:73) E · d s = − d Φ dt = − ddt (cid:90) b · d F (12)and the right-hand side is the exchange of magnetic flux inthe reconnection process within the typical time dt = τ rec .This time is not necessarily the same as the anomalous colli-sion time. The magnetic flux is given by ∆Φ ≈ π b ∆ / k (cid:107) . Theline integral over the boundary of the reconnection site be-comes ≈ π E z / k (cid:107) + ∆ δ E x . Under ideally symmetric condi-tions the second term would vanish because the two contri-butions of the x integration would cancel out. If some asym-metry is retained then a finite component δ E x arises. Takenthese together yields dimensionally (not caring for the signs)4 π E z / k (cid:107) + ∆ δ E x ≈ π b ∆ / k (cid:107) τ rec (13)Neglecting the small second term on the left then gives a sim-ple order of magnitude estimate of the reconnection electricfield E z ≈ b ∆ τ rec (14)which could have been guessed from the beginning. Thiscontains the reconnection time τ rec which so far is unde-termined. It can be taken for instance as the above derived anomalous collision time τ an = ν − an . Below we derive anothercharacteristic time. Which one has to be chosen, cannot de-cided from these theoretical order of magnitude estimates. Itis either due to observation or numerical simulations.The small additional term 2 ∆ δ E x = − U is a potential fieldproduced by a possibly present asymmetry between the orig-inal current sheets or some gradient in the particle density.Such a gradient can be produced, if a substantial part of theelectron component in the gap is accelerated away along themain field, causing a dilution of plasma in the exhaust. Beingperpendicular to the magnetic fields B and b it leads to weakshear motions and circulation of the electrons inside the gap-exhaust region, which should observationally be detectable. In the above we have made use of the notion of reconnectiontime τ rec . Here we attempt a clarification of this time. Top-side reconnection will not be stationary. It should vary on thetime scale of the kinetic Alfvén frequency respectively mov-ing together with the latter along the magnetic field. This mo-tion should mainly be upward since causality requires thatthe wave transports information back upward with the up-ward moving electrons in the downward current region. Itwill thus be modulated and lead to quasi-periodic acceler-ation and generate medium energy electron bursts ejectedfrom the local electron exhaust reconnection region along thesheet current magnetic field. These bursts flow perpendicularto the ambient field, start gyrating and immediately becomescattered along the ambient field spiralling mainly upwardinto the weak ambient field region. Their pitch-angle distri-bution should obey a well defined downward loss-cone.With the above estimate of the anomalous resistivity in thiskind of reconnection, we can proceed asking for the typicalreconnection time scale. For this purpose we return to Poynt-ing’s full theorem and take its variation with respect to thestationary state, indexing the latter with 0 while keeping theslow perpendicular velocity V ⊥ fixed but varying the resis-tivity. We need to express the parallel current through theresistivity. This can be done via the electric field E (cid:107) to obtain J (cid:107) = η − E (cid:107) = η − b V ⊥ (15)This procedure, after some straightforward and simple alge-bra and rearranging, leads to the following expressions d ( δ b ) dt ≡ (cid:18) ∂∂ t + V ⊥ ∇ ⊥ (cid:19) ( δ b ) = − µ J (cid:107) δη (16) δη = − V ⊥ µ J (cid:107) ∆ ( δ b ) (17)and we obtain dimensionally for the typical time of recon-nection τ rec ∼ ∆ V ⊥ V ⊥ ≈ ∂ω/∂ k ⊥ , one obtains τ rec ≈ ∆ V A k ⊥ k (cid:107) (cid:16) + k ⊥ λ e (cid:17) k ⊥ λ e (19)a time the length of which depends essentially on the spac-ing of the current sheets. Since V A is large, there will be abalance between the spacing and the domain of the kineticAlfvén wave spectrum which allows reconnection to occurin the topside. Let the vertical topside width be L z and theAlfvén time τ A = L z / V A then we have the condition τ rec τ A ≈ ∆ L z k ⊥ k (cid:107) (cid:16) + k ⊥ λ e (cid:17) k ⊥ λ e < ∆ of the sheets, meaning that √ ∆ L z < k (cid:107) k ⊥ = λ ⊥ λ (cid:107) (cid:28) ∆ > r ce . Both conditions are easily satisfied. In the present letter we propose that reconnection might oc-cur not only in given current sheets but also in the topsideionosphere-magnetosphere auroral transition region wherethe main magnetic field is very strong, almost vertical, anddirectly connects to the tail reconnection region. It serves asa guide for any particle flow exchange between the topsideionosphere and the tail plasma sheet, exchange between lowfrequency electromagnetic waves (in our case kinetic Alfvénwaves) trapped in flux tubes and the accompanying field-aligned current sheets, and ultimately as an inhibitor for thefield-aligned parallel current sheets to merge. This enablesreconnection in the gap between the current sheets betweenthe oppositely directed magnetic field of the sheets respec-tively the kinetic Alfvén wave magnetic fields.Dealing with reconnection, one is not primarily interestedin the change of magnetic topology but in energy transfor-mation from magnetic into kinetic, di ff usion of plasma andmagnetic field across the reconnection region, generation ofelectric fields, and ultimately selective particle accelerationas these are the observed e ff ects. The generality of recon-nection is not the best argument. The decades old claim thatreconnection converts magnetic energy into mechanical en-ergy is no fundamental insight; in all processes involving re-connection, the main energy is stored in the basic mechanical motion and by no means in the magnetic field. This motion,convection in inhomogeneous media with boundaries, likethe magnetotail or the magnetopause, or turbulence neces-sarily produces currents and transports magnetic fields to letthem get into contact. The amount of energy released by re-connection is in all cases just the minor electromagnetic part,a fraction of the mechanical energy.Topside reconnection is expected predominantly in thedownward current region, which observationally seems to behighly structured, consisting of several adjacent parallel cur-rent sheets. Similar conditions may also occur in the upwardcurrent region though no such structuring is obvious fromobservations. If it exist, then the physics will be similar. Wehave shown that topside reconnection is possible, generates aelongated field-aligned regions (exhausts) where the fields ofparallel current sheets merge, anomalous collisions are gen-erated, energy is exchanged and dissipated, and most impor-tant a first order reconnection electric field E rec is producedin the exhaust along the ambient magnetic field but restrictedto the gap region between the current sheets. This field iscapable of accelerating electrons along the main field, as ismost desired by all auroral physics. Here it comes out as anatural result of topside reconnection. Topside reconnectiongenerates parallel electron beams, it lifts the escaping elec-trons in the exhaust into an elevated parallel energy level.These beams then cause a wealth of auroral e ff ects in the en-vironment and when impinging onto the upper ionosphere.Acceleration of electrons by the reconnection electric fieldleaves behind an electron depleted exhaust mainly contain-ing only an anisotropic electron component whose pitch an-gle distribution peaks at perpendicular energies.It is instructive to briefly inspect Fig. 3. It shows thedownward (upper panel) and upward (lower panel) electronfluxes. In addition to the temporally / spatially highly struc-tured fluxes, still obeying the spatial di ff erences between thedownward and upward current regions imposed by the tail-source of the downward fluxes, resulting from variations intail-reconnection, or several tail-reconnection sites, one oc-casionally observes the simultaneous presence of upward anddownward fluxes in the downward current region. One partic-ular case it at t ≈
60 s. The upward electron fluxes maximizebelow ∼ . ∼ . ffi ffi cient.These conditions allow for the plasma to become an emitter(Twiss, 1958; Schneider, 1959; Gaponov, 1959) by the elec-tron cyclotron maser mechanism (Wu & Lee, 1979) basedon a loss-cone distribution (Louarn & Le Quéau, 1996). Itrequires weakly relativistic electrons (see Melrose, 1989;Treumann, 2006, for reviews) and a low density electronbackground embedded into a strong field. It nicely comesup for the weak auroral kilometric background radiation butfail explaining the intense narrow band observed and driftingemission seen in panel d of Fig. 2.To explain the latter, in earlier work we referred to elec-tron hole formation (Pottelette & Treumann, 2005; Treumannet al., 2011b). Hole models favourably apply to electron de-pleted exhausts in topside reconnection where densities be-come low (see, e.g. Treumann & Baumjohann, 2013) andthe remaining trapped electron component maximizes at per-pendicularly speeds having large anisotropy. Intense narrowband drifting emissions in the frequency range 300-600 kHzmay be a signature of topside reconnection in the strong mainauroral field. They were originally attributed to Debye scaleelectrostatic electron holes (Ergun et al., 1998b; Potteletteet al., 1999) observed by Viking (de Feraudy et al., 1987)and FAST (Carlson et al., 1998; Ergun et al., 1998a; Pot-telette & Treumann, 2005) but are to small-scale for radiationsources. Topside reconnection exhausts instead have dimen-sions along the magnetic field of half a kinetic Alfvén lengthand transverse scales of few ion inertial lengths λ i or ∼ λ e .Such scales can host and amplify one or more radiation wavelengths.Of course, details of this process should be developed bothanalytically as far as possible, and by numerical simulations.If confirmed, this mechanism would also map to any astro-physical moderately or strongly magnetized object with ap-propriate modification.The present qualitative considerations which we spicedwith a few simple estimates based on energy conservationarguments just propose that reconnection in the topside au-roral ionosphere is a process which has so far been missedand probably is that mechanism which releases the largestamount of so-called magnetically stored energy availableand from the smallest spatial regions. Reconnection in muchweaker fields like in turbulence and broad current sheetswill be substantially less e ffi cient because of the weaknessof the reconnecting magnetic fields. Nevertheless in verylarge extended systems with reconnection proceeding on themicroscales (Treumann & Baumjohann, 2015) with the to-tal number of reconnection regions very large, the emissionmeasure is large as well, and radiation from reconnectionmay become a non-negligible signature even in weak fields.However, in very strong fields like those in magnetized plan-ets and magnetized stars (predominantly neutron stars, whitedwarfs but also including outer atmospheres of magnetizedstars like the sun) reconnection following our argumentationmay be more important than so far assumed. Acknowledgments
This work was part of a brief Visiting Scientist Programme atthe International Space Science Institute Bern. RT acknowl-edges the interest of the ISSI directorate as well as the gen-erous hospitality of the ISSI sta ff , in particular the assistanceof the librarians Andrea Fischer and Irmela Schweitzer, andthe Systems Administrator Saliba F. Saliba. We acknowledgediscussions with R. Nakamura, and Y. Narita. RT acknowl-edges the cooperation with R. Pottelette two decades ago onthe data reduction and the radiation and electron hole prob-lems. References
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