Formation of the small-scale structure of auroral electron precipitations

Author’s Accepted Manuscript Formationof the small-scale structure of auroral electron precipitations A.P. Kropotkin

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To appear in: Journal of Atmospheric and Solar-Terrestrial Physics Received date: 2 February 2016 Revised date: 16 August 2016 Accepted date: 18 August 2016 Cite this article as: A.P. Kropotkin, Formationof the small-scale structure of auroral electron precipitations, Journal of Atmospheric and Solar-Terrestrial Physics, http://dx.doi.org/10.1016/j.jastp.2016.08.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Formation of the small-scale structure of auroral electron precipitations A.P. Kropotkin Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991 Russia [email protected]

Abstract This paper is aimed at physical causes of the small-scale transverse structure in the flows of auroral electrons, generating the corresponding small-scale structure of discrete auroras. The parallel electric field existing in the lower part of the auroral magnetosphere, in the auroral cavity region, in the presence of a strong upward field-aligned current, accelerates magnetospheric electrons to energies of

1  10 keV. The flow of these particles while maintaining the high density of the field-

aligned current, produces a current-driven instability, which generates Alfvénic turbulence at short perpendicular wavelengths  1 km. These short-wavelength inertial Alfvén disturbances possess a nonzero parallell electric field, which modulates the electron flow velocity. The modulation occurring at high altitudes  104 km leads to a nonlinear effect of formation of strong density peaks at low altitudes of electron precipitation. The transverse, horizontal scales of the corresponding electron flow structure coincide with the small scales of the Alfvénic turbulence; and this structuring leads to non-uniformities in the auroral luminosity on the same scales, i.e., to smallscale structure of discrete auroras.

Keywords: Auroral electrons; velocity modulation; Alfvénic turbulence; nonlinear flow evolution; aurora; small-scale structure

1. INTRODUCTION After decades of intense research, plasma dynamics of the auroral region is still one of the most difficult and interesting problems in the physics of the magnetosphere. Of course, there is already a number of well-founded ideas, based on the results of satellite and ground-based measurements. During the last two decades, in a number of excellent experiments, a large amount of data was collected (e.g. [Paschmann et al., 2003; McFadden et al., 1999a,b; Chaston and Seki, 2010; Chaston et al., 2010, 2011; Ergun et al., 2000]), concerning the particle and fields environment in the auroral upper ionosphere – lower magnetosphere region. As the authors of publications point out, those results basically confirmed the general concepts established earlier and mainly concerning the large-scale structure of the auroral plasma system. In the disturbed magnetosphere, field-aligned flows of energetic electrons precipitating from the magnetosphere, with energies

1  10 keV, form “inverted- V ” structures in the latitudinal direction: electrons of

the quasi-monoenergetic flow have the maximum energy at the central latitude, with decrease of energy at higher and lower latitudes. The latitudinal scale of such a structure is of the order of hundreds of kilometers. The electron flows in the “inverted- V ’s” carry a field-aligned current of high density: those are the most powerful field-aligned currents involved in the three-dimensional current system of the disturbed magnetosphere. However the experiments mentioned above, were mostly aimed at the further study of the auroral physics in detail, in the wider spatial range and on the fine spatial and temporal scale. Correspondingly, for their interpretation a number of more or less sophisticated models were proposed. In many of them much resemblance of results was achieved in relation to the observational data concerning the structuring of electric and magnetic fields, of particle distribution functions, and of auroral forms. In particular, auroral observations show that a the heights of the glow,

100 km, the flows

of energetic electrons that generate auroras, are highly horizontally structured: they generate small-

scale structure of discrete auroras, whose horizontal scales range from a few kilometers to tens of meters [Borovsky, 1993, Sandahl et al, 2008, Kozelov and Golovchanskaya, 2010]. An important feature of discrete auroral forms is that their spatial spectrum is close to that of transverse scales of simultaneously observed Alfvénic turbulence [Stasiewicz et al, 2000a, Golovchanskaya et al., 2011]. Such electromagnetic wave turbulence is observed aboard spacecraft in the auroral region above the auroras, in the upper ionosphere - lower magnetosphere region, at altitudes  103 km [Gurnett et al, 1984, Lindqvist and Marklund, 1990, Stasiewicz et al, 2000a, Ergun et al, 1998]. At relatively high frequencies (in the satellite frame) corresponding to large transverse wave numbers, the Alfvénic disturbances are actually inertial Alfvén waves [Goertz, 1984, Stasiewicz et al, 2000b]. They have a nonzero magnetic-field-aligned component of the electric field, so that they interact effectively with parallel electron flows. The origin of large-amplitude small-scale Alfvén waves in the regions occupied by auroral field-aligned currents is a matter of many recent studies. One of the most promising approaches is the premise that such waves might be produced by self-consistent magnetosphere-ionosphere (M-I) coupling [Streltsov and Lotko, 2004; Streltsov and Karlsson, 2008; Russell et al, 2013]. Numerical experiments have demonstrated that waves resembling observations are produced by a simulated system’s response to large-scale field-aligned currents via the ionospheric feedback mechanism (IFM). While this mechanism may really be responsible for generation of large-amplitude smallscale Alfvén waves and associated current structuring in the auroral upper ionosphere –lower magnetosphere region, we propose in this paper that those small-scale inertial Alfvén waves which are eventually responsible for the strong structuring of energetic auroral electrons, should be generated by current-driven instability above that region, in the very low-beta plasma typical for auroral cavity environment [Chen et al, 2013] where the strong field-aligned current is carried by electrons accelerated at even higher altitudes.

Another approach to the problem was proposed in the paper [Chaston and Seki, 2010]. The modeling there is based on the equations of the single-fluid MHD theory for a plasma with a fieldaligned current initially distributed smoothly over a large transverse scale. A uniform magnetic field is assumed, so that the influence of magnetic mirroring is excluded in principle. In such a model evolution occurs, leading to transverse structuring of the field-aligned current, based on Alfvénic turbulence, and correspondingly, of the vertical energy flux. However, although there is similarity of these structures with the observed auroral forms [Chaston et al., 2010, 2011], the real physics might be different. In fact, in “inverted-V” regions with very intense upward field-aligned current, the small-scale precipitation patterns are generated specifically by the structured quasi-monoenergetic electron beams. The plasma model must fundamentally be kinetic: the electron distribution function is a beam (coming from outside) on the cold isotropic background. Evolution of such a system with a beam is described by equations different from the hydrodynamic equation set adopted in the paper [Chaston and Seki, 2010]. Of course, in such a fluid model there also exists a possibility of nonlinear evolution processes in the plasma of the auroral upper ionosphere - lower magnetosphere, resulting in transverse structuring which is just shown in [Chaston and Seki, 2010]. However, in our opinion, in the first place attention should be paid to the nonlinear process of that strong structuring at low altitudes, which is initiated by weak variations of an initially wide electron flow earlier accelerated at higher altitudes, where it should basically be formed in order to provide a strong field-aligned current observed in “inverted V’s”. We would like to emphasize here an associated fundamental issue which, in our opinion, has not been paid enough attention in those recent studies. It was pointed out back in the 80-ties that during geomagnetically disturbed times, very strong upward field-aligned currents are involved in the three-dimensional magnetosphere-ionosphere current system (“substorm current wedge”). Such a current cannot be maintained by the upward ion flow from the ionosphere: it proves to be insufficient. It follows that the current should be carried by magnetospheric electrons. However in

normal conditions, even with the loss cone totally filled-up, the flow of hot magnetospheric electrons is also insufficient (see e.g. [Kivelson and Russell, 1995]). To improve the situation, the loss cone must be widened. And this is only possible if an electric field appears along the magnetic field lines, accelerating electrons down the field lines. Importantly, the localized potential drops appearing in the region of upper ionosphere – lower magnetosphere, which have been often included in simulations, actually cannot serve the aim. Since the electrons, in their bounce motion, are mirroring in the converging magnetic field, the sufficient number of electrons can only be collected at higher altitudes, by means of the loss cone widening over there. The presence of parallel electric fields in the inertial Alfvén waves has led some researchers to seek a direct causal link between the waves and the flow of auroral electrons in the belief that their acceleration up to 110 keV can occur under the action of parallel fields of the inertial Alfvén waves (e.g., [Thompson and Lysak, 1996, Lysak, 1998, Lysak and Song, 2003, Wu and Chao, 2004]). We do not generally deny such a possibility. But (1) it certainly may be that those inertial Alfvén waves which demonstrate similarity of the spectra to those of auroral forms are not those waves which are thought to be responsible for electron acceleration: mesoscale mechanisms that ensure a significant fraction of the auroral acceleration are separated from the small-scale acceleration, due to dispersive Alfvén waves, which contribute to the transverse wave structuring similar to precipitation structuring; (2) we believe that generally the waves play a key role not in the basic mechanism of electron acceleration up to the auroral energies but in the mechanism of the horizontal structuring which is observed at low altitudes where the auroral luminescence is excited. As to the basic mechanism of electron acceleration, it was pointed out a long time ago that the electric potential drop distributed along the magnetic field line, provides acceleration of magnetospheric electrons down toward the ionosphere. The flows of accelerated electrons are really observed in the auroral upper atmosphere as the well-known “inverted V” structures. These electrons form quasi-monoenergetic beams there. It has been shown that the current density jI

eneI vI is nearly proportional to the overall potential drop U , from ionosphere to the equator (

neI and vI are the electron number density and parallel velocity, respectively, at the ionospheric level), [Knight 1973]. This result is known as the “effective Ohm’s law” jI

 KU ;K 

nM e2 2 mTM

with the subscript M relating to the magnetospheric equatorial region. However, this is only an approximate relation, with a limited range of validity; a number of refinements have been later proposed [Antonova and Tverskoy, 1975; Whipple, 1977; Chiu and Schulz, 1978; Stern, 1981]. Note also that basically this is a kinetic effect, it appears in the collisionless kinetic theory. If we turn to the fluid theory as e.g. in [Chaston and Seki, 2010], there is no reason to build up a finite conductivity and related diffusion of the magnetic field on this basis. True, the particle distributions which arise, may prove to be unstable relative to plasma waves generation, and this can lead to plasma wave turbulence and associated anomalous resistivity, but this is another story. There is a number of papers where such effects, being localized and not global over a field line, have been analyzed (e.g. [Lysak and Dum, 1983; Lysak, 1990]). And this approach might also be quite fruitful in understanding the electromagnetic structures in the upper ionosphere – lower magnetosphere. Anyway we believe that the structuring which arises as a result of those corresponding simulations, should be looked at as complimentary to the effect analyzed in this paper, at least for situations with the strongest upward field-aligned currents as pointed out above. In our approach, we turn to the real physical nature of the large-scale distributed potential drop itself. That nature was identified in a series of papers [Kropotkin and Martyanov, 1985, 1989, Kropotkin, 1985, 1986]. It was done along the following lines. (1) The mechanism of charge separation acting in the near-equatorial region and forming the substorm current wedge, thus providing a source of a strong upward field-aligned current, was identified [Kropotkin and Martyanov 1989, Fridman, 1994, Ergun et al, 2000, Schriver, 1999]. When the situation may be described in the terms of a quasi-state condition, the field-aligned currents are basically determined by the feature of axial asymmetry in the hot plasma distribution in the ring current region. This

asymmetry appears and increases during substorm plasma injections from the plasma sheet. The

1 well-known Vasiliunas – Tverskoy formula, jI  c  n W p  [Vasiliunas 1970; Tverskoy, 2 1982] provides the field-aligned current density in this case. Here W  

ds is the volume of a flux B

tube with a unit magnetic flux, p is the plasma pressure, and n is the normal to the ionospheric surface. (2) Formation of difference was analyzed between the distribution anisotropies of hot magnetospheric ions and electrons, in the presence, along with the mirror force, of a large-scale distributed parallel electric field, leading to existence of a steady-state upward field-aligned current, governed by the quasi-neutrality equation; the structure is determined by action of both the electron inertia and the mirror force. (3) The presence, at the foot of a flux tube, of cold and dense ionospheric plasma prevents the parallel electric field from the magnetosphere penetrate onto the ionospheric heights. The ionosphere proves to be separated from the electron acceleration region where that field acts, with an electric double layer having a small scale in the magnetic field direction, comparable to the Debye length. (4) Accordingly, on those flux tubes where a region of electron acceleration is present, in that region over the double layer the plasma density is very low, on the orders of magnitude lower than that at the same altitudes in the surrounding plasma of the upper ionosphere but outside that region. From the observational data, this feature has long been known as the “auroral cavity”. The time-dependent evolution leading to the situation described above, has been also analyzed [Kropotkin and Martyanov 1989, Fridman, 1994, Ergun et al, 2000, Schriver, 1999].The beginning of charge separation in the equatorial region of a flux tube, is followed, in the case of upward field-aligned current, by appearance of a nonlinear collisionless “shock”, i.e. an electrostatic potential jump propagating downwards and separating the domains of dense cold ionospheric plasma and rarified hot magnetospheric plasma. This propagation can stop, with formation of a steady-state auroral cavity, at the time when the inflow of electron charge at the equator becomes balanced with the outflow by means of auroral electron precipitation. This

however may be also accompanied by formation of a number of various plasma discontinuities being in relative motion along the field tube. It is worth noting that actually all the features of the above theoretical scenario, which relate to the auroral upper ionosphere – lower magnetosphere, have been really observed (e.g. [McFadden et al., 1999a, Paschmann et al., 2003]). Moreover, as it has been mentioned above, observations provide also many additional features which have been interpreted in later simulation attempts (e.g. [Chaston and Seki, 2010; Ergun et al., 2000]). In this paper we propose a new scenario of processes which cause such an important feature as the small-scale transverse structuring of the auroral electron flows, corresponding to small-scale structure of discrete auroras. The parallel electric field existing in the lower part of the auroral magnetosphere, at altitudes  104 km, in the auroral cavity region, in the presence of a strong upward field-aligned current, accelerates magnetospheric electrons to energies of

1  10 keV. The

flow of these particles while maintaining the high density of the field-aligned current, produces there a current-driven instability, which generates Alfvénic turbulence at short perpendicular wavelengths (Sec. 2). These short-wavelength inertial Alfvén disturbances possess a nonzero parallell electric field, which modulates the electron flow velocity (Sec. 3). The relatively weak modulation occurring at high altitudes  104 km leads to a nonlinear effect of formation of strong density peaks at low altitudes of electron precipitation (Secs. 4 and 5-6). The transverse, horizontal scales of the corresponding electron flow structure coincide with the small scales of the Alfvénic turbulence as discussed in Secs. 7 and 8.

2. CURRENT-DRIVEN INSTABILITY It is known that in rarefied cold plasma, with   me / mi , i.e. vA  vTe , vTi (the symbols have their usual meaning), the effect of electron inertia manifests itself in dispersion of low-frequency

waves on the Alfvénic branch (AW) with small transverse sizes, so that (e.g., [Stasiewicz et al, 2000b])

2 

vA2 k z2 1  k2 re2

(1)

( re  c /  pe is the electron inertial length). In such waves, unlike the long-wavelength Alfvénic perturbations, a nonzero parallel electric field appears, so that E E

At k re



k k re2 1  k2 re2

.

1 we obtain E   k / k  E and the wave becomes electrostatic [Stasiewicz et al, 2000b,

pp. 430-437]. The low value of the electron density n in the auroral cavity leads to a large value of the electron inertial length re , and the inertial dispersion effect in AW manifests itself already beginning with such transverse scales,

 pe  4 ne2 / me

re . Just to get an idea, at n 10 cm-3 we have

1.5 105 (c-1); re  c / 0e = 2 km.

Thus the small-scale structure of the Alfvénic turbulence in the auroral cavity, with transverse scales  re

1 km, has the features of electrostatic perturbations (a “hybrid” of AW and

longitudinal plasma waves). Based on Eq. (1), their spectrum is approximately given by the formula



B0 c 4 nmi

4 ne2 k z eB0 mi k z k z   0i 0 e , me k mi c me k k

0i ,0e are the ion and electron gyro-frequencies, correspondingly. The above formulas and estimates relate to a current-free plasma. In the presence of an intense field-aligned current in the plasma, they are considerably modified. A possibility of current-

driven instability appears, i.e. excitation of inertial Alfvén waves by means of an electron flow through ionic background. The current-driven instability of inertial Alfvén waves in a plasma of very low pressure, particularly in the auroral cavity plasma, was studied in [Chen et al, 2013]. If the velocity v0 of magnetic-field-aligned drift of electrons carrying the current, is high enough,

v0 v0c   vA vA

1  k  1  3 k  2 2  e

2 2  e

k e

/ 4

,

i

i k2 vTi2 nmi vTi2 B02 4 nmi vTi2 vTi2 (here     1,i  2 ,i  /   2 ;Q  me / mi ), then the 2Q k2 e2 ci 2 8 B02 vA solution of the dispersion equation shows instability. The frequency r of the wave and its growth rate i may be presented as follows:

r k vA

i k vA





k2 e2 v0 , 1  k2 e2 v A k2 e2 v02 / vA2  1  k2 e2 1  3 k2 e2 / 4  1  k2 e2

.

(2)

Analysis of these formulas reveals those values of the transverse wave number at which the growth rate is at maximum. For a fixed value of k , the maximum is reached at k e

2 . To estimate the

value of k at which the absolute maximum is reached, and the very magnitude of that maximum, we must bear in mind that for large k e we have r

k v0 . But for low-frequency waves under

consideration, this frequency should be much lower than the ion gyro-frequency i , and so

k v  0

2

i2 . But on the other hand, for instability the inequality v0  vA should be valid. Thus,

the maximum possible value of the parallel wave number, which corresponds to the maximum of

i for any given value of k e , may be estimated as km

0.3i / vA .

(3)

Accordingly, for large v0 / vA , the growth rate reaches values of the order  0.1  0.3 i [Chen et al, 2013]. In this paper we use typical parameters of the acceleration region of auroral electrons mainly following [Chen et al., 2013]. At the heights

1  2RE , the observed highest current density is

about j 1 μA / m2 (as compared to j 10 μA / m2 at the ionospheric height), and the magnetic field, plasma density and temperature in the auroral cavity are of the order of the values B0 G, n0

1 cm-3 and Te

0.01

300 eV, respectively. The latter figures roughly correspond to those in the

former FAST work (e.g. McFadden et al, [1999a, b]). This gives the values of the ion gyrofrequency ci vA

2 102 s-1, of the electron plasma parameter e

2 109 cm/s and of the parallel electron velocity v0 / vA

105 , of the Alfvén speed

1.7  1 (correspondingly, the electron

energy in the flow is mv02 / 2 10 keV; note that this is about one order of magnitude higher than what is presented in several FAST results [McFadden et al, 1999b], this presumably only means that the measurements relate not to the cases with the largest upward current density; earlier measurements reviewed in [Lin and Hoffman, 1982] indicate that there are narrow bursts of intense electron precipitation fluxes imbedded within some (not all!) inverted-V’s, which have the proper characteristics to cause discrete auroral arcs in the atmosphere, and in those bursts the electron energy in the spectrum peak is often about 10 keV). Note also that the ion plasma parameter is

i

3 106 , for Ti

30 eV [McFadden et al, 1999b], and the condition i

instability theory [Chen et al., 2013] is valid. With vA

km

me / mi of the

2 109 cm/s we obtain, according to (3),

3 108 cm-1, so that the parallel, altitudinal scale of waves excited by the instability, should be

about a thousand kilometers. As to the frequency of the excited waves, according to (2) and (3), at ke

rm

k mv0

3 108  4 109 102 s-1 so that  rm  rm / 2

2 we have

20 s-1. Thus, as we see, predominant

wave frequencies of the Alfvénic turbulence should be in the range of tens of hertz. Note that this

range can differ significantly from that observed aboard satellites, the order of several hertz [Stasiewicz et al, 2000b], since quite different heights are meant: we consider here interaction with the electron beam and wave excitation which occur at much higher altitudes than the height of the satellite observations, e.g. onboard FAST (altitude range 400 – 4000 km). In addition, this frequency is determined based on satellite measurements in a very indirect way: it is much less than that oscillation frequency which is directly measured and is basically determined by the transverse spatial rather than temporal structure. 3. MODULATION OF THE ELECTRON BEAM Fast particles of the electron beam passing through the region occupied by the Alfvénic turbulence (which is actually generated by the instability of that same electron beam), change their energy in random parallel electric fields of that turbulence. Resonant interaction of the electron beam with waves leads to instability (sec. 2), i.e. to generation of waves propagating down to the ionosphere, and to the particle energy loss. In the emerging wave turbulence, as the experimental data processing shows [Golovchanskaya and Maltsev, 2004], the energy flux is really directed downward, i.e. the energy flux dominates in those waves with the group velocity directed downward. However, due to nonlinear interactions of the turbulent waves and to the presence of the ionospheric Alfvénic resonator [Trakhtengertz and Feldstein, 1984], backward waves must also be present in the emerging turbulence. In addition to the energy loss in resonant interactions, the electrons take part in non-resonant interaction with sign-alternating electric field of the wave turbulence. As a result, in rough terms, the picture looks as follows. An electron on its field-aligned trajectory experiences a sequence of sign-alternating random impacts, each one over a segment with length of the order of the Alfvénic turbulence correlation length. This correlation length should be the order of the characteristic parallel wavelength of the Alfvén waves,   2 / k , if it is assumed that the waves form a set of wave packets with random phases, in a wide frequency range. The number of random impacts on the length of the acceleration region d is of the order of n

d / .

Accordingly, if on a separate segment of its trajectory, the electron changes its energy by an amount

 eE  

eE  , then the total change of its energy is estimated as 

d /   eE  d . Then

for the variation of the beam velocity we obtain v

 / mv0

eE  d / mv0 .

Note that henceforth we assume, in a quite approximate manner, that the bulk flow velocity v0 remains nearly the same throughout the region despite the action of magnetic mirroring. This is valid if the parallel motion energy much exceeds that of perpendicular motion, everywhere in that region, due to acceleration in the background large-scale electrostatic field, so that the flow remains nearly monoenergetic. The relative velocity variation is

v v0

eE  d mv02

.

(4)

As mentioned in Sec. 2, short-wave perturbations, k re  1 , are nearly electrostatic, E / E

k / k .

For given values of re ,k ,E , the parallel field is proportional to the parallel wave number, i.e. E   const . Then all the values  ,v,v / v0 are proportional to  1/2 .

For a rough estimate, based on observations, we may set E

100 mV/m, 

1 km. The

characteristic scale of the electron acceleration region in the lower auroral magnetosphere may be set equal to d equal to  0

104 km. The characteristic value of the auroral electron energy is set as previously

10 keV, i.e. v0

6 109 cm/s. Regarding the value of the parallel wave number (or the

parallel correlation length), there are no certain observational data, as well as for the relative value of E . In Sec. 2 we however pointed out that the mostly growing waves have the parallel scale



100 km. Inserting the earlier obtained parameter estimates into Eq. (4),we get

v v0

eE  d 2 0

mv



100   2 107 10

eE  d 2 0

mv



3 1/2

 

(5)

102 e  eV  / 2 104  eV  101.

4. FORMATION OF NONUNIFORMITIES: ASSESSMENT We now turn to the effect of “caustic” which explains the growth of inhomogeneities in the electron flow in the auroral magnetic flux tube, under the influence of speed variations at the initial level where the density is uniform. It is easy to estimate the characteristic upward distance from that level at which the density intensifications are formed. Using the above notations, we find an estimate of the time which a rapid flow with velocity v0  v needs to catch up a slow one, with velocity v0 , at a distance d :

T

d

1 1  v0  v v0

d v . v02

In this way we obtain an estimate of possible characteristic period for modulation of the flow at the initial level, T

2



d  v . v02

(6)

Clearly, this is an estimate of the maximum possible period T at which the effect can exist. At lower values of the oscillation period T in a wave modulating the speed at the initial level, the first density peak will appear at a smaller distance from the initial level than d , and several times peaks appear again along the field line, including that at the ionospheric level, h 100 km. For the above estimated values, taking Eq. (5) into account, we obtain Tmax 

d  v 2 104  0.1  3 102 s. 2 4 v0 6 10

So the minimum possible value of the angular frequency is min  2 / Tmax

102 s-1.

As it is seen from the estimates presented at the end of Sec. 2, the frequencies of waves excited by the current-driven instability, are in this same range. It is also seen that the limiting frequency for the “caustic” effect is   2 relative change in speed was obtained above, and at E / E

v v0

eE

v02 . The d  v

k / k it can be rewritten in the form:

k

2 d / k

k

mv02

.

Using the resonance condition   k v0 , we can now estimate the parallel scale and the parallel field magnitude corresponding to this limiting case, without referring to results of observations. We have

k v0

2 v02 d v

k 2 v0 k , mv02  d k eE 2 d

so that

kd

 2 mv02 k    eE  

If, as before, we set E  100 mV/m, k  2 /  eE / k

2/3

.

6 103 m-1, mv02 / 2  10 keV, then

17 eV, and thus k d 100 . Accordingly, at d  2 104 km, we obtain

  2 / k

1000 km.

This estimate is consistent with that previously obtained at the end of Sec. 2, based on entirely different reasons, from the condition of maximum growth rate of the current-driven instability. And as we see, the condition d / 

1 necessary for our scenario of stochastic

acceleration (deceleration) of the beam, proves to be valid. 5. KINETIC THEORY

The above considerations concerning the formation of strong density irregularities in the flows of auroral electrons are readily presented in a quantitative form. To do this, it is necessary to use the kinetic theory. We are limiting the approach here to the simplest version of this theory, applying the free flow approximation to electrons. This is a very rough assumption which ignores the fact that the formation of electron bunches must be accompanied by formation of an uncompensated electric charge. The relevant discussion is postponed until the final Section 8. In a simplified version such an analysis was carried out long ago in application to the solar wind flow [Veselovsky, 1979, 1980]. The initial velocity distribution function was proposed deltafunction-like. The appearance of density peaks was analyzed, being associated with the emergence of multi-flow motion. Here we assume that the distribution function has a finite width because of thermal velocity spread of particles. We set x a coordinate along a field line, and v - the electron velocity component in that direction. We assume that the electron flow has a bulk velocity much greater than the thermal parallel and perpendicular velocity components everywhere (despite the growth of the latter in the converging magnetic field). Then the distribution function f  x, v, t  for the free flow along a field line obeys the Liouville equation f f v 0. t x

Solution has the form f  x, v, t   F  , v  , where   t  x / v , so

(7) f f  F' ,   F' / v , and the t x

function F can be defined by the boundary condition, for example in the form f  x  0, v, t  

2  1 exp  2  v  v0  v1 sin t   .    

N0

Here N 0 is the particle number density at the level x  0 ,  is the effective width of the velocity distribution, v0 is the central value of the directed velocity, v1 is the velocity modulation amplitude; we will consider it small, v1

v0 .

Introduce   v  v0 , then

1  v

   x and sin   t  x / v    sin   t  .  v 1   / v         0 0    v0 1    v0 

1

The solution of the Liouville equation (7) may be presented in the form

  1 f  x, v, t   exp  2    N0

     x   v1 sin    t      v 1   / v    0 0     

2

  . 

(8)

Let t  x / v0 , then   1 f  x,  , t  x / v0   exp  2     N0

  x  / v0     v1 sin     v0 1   / v0   

2

  .  

 x  / v0  x  / v0 . Introduce a Let us initially assume x so small that sin     v0 1   / v0  v0 1   / v0  dimensionless value    /  . Then 2    2 v1 x   f  exp  1  2   . v0 1   / v0         

N0

(9)

Consider the interval of x values, wherein while estimating the width of the distribution over  , 1

  v1 x 1  2  , the additive term  / v0  may be neglected compared to the unit. v 1   / v      0 0   Introduce xm 

v02 . v1

(10)

If we drop out the said addition, the width of the distribution over  is given by the expression: v x 1 1 2 v0

1

v x v  1  1 2 m  1 2  x  xm  v0 v0

1

v02 .  v1 x  xm

And if

v02  v1 x  xm v0

1 , it is really possible to neglect the specified additive term. Thus, we

obtain a double inequality as a condition for the specified additive term to be reasonably discarded, but at the same time the width of the distribution to be large:

 v0

v1 x  xm v02

1.

(11)

The left inequality here just provides a large value of density N as compared to N 0 , defined as the integral of f over  . In fact, we have already used one more approximation which imposes a limit on the x  xm value. Since the width of distribution over  is given by an expression 

corresponding width of distribution over  is equal to 



v02 , the v1 x  xm

v02 . Therefore, at the v1 x  xm

 x  x  / v0  v0 boundary of this distribution we have sin    sin    v0 1   / v0   v0 v1 x  xm

The small value of the argument,

that x  xm

x  x  xm v1

1 at x  xm 

  x    sin   . x  x v m 1   

v02 , that we have assumed, implies v1

v02  . Therefore, the double inequality (11) should be replaced by the following: v12 

 v1

v1 x  xm v02

1.

(12)

As a result we obtain the distribution (8) in the form

  v  x  x  2  f  exp   1 2 m   2  . v0      N0

(13)

Integration of the distribution (13) over velocity v  v0   in infinite limits provides the density

N  x  at the point x  v0t : N  x   N0

v02 . v1 x  xm

It follows that in the vicinity of the point x  xm 

(14)

v02 v , at time values near t  tm  xm / v0  0 , v1 v1

the density N  x  increases greatly, and when the left condition (12) is valid, it becomes much greater than the initial density N 0 (since  / v1

 / v0 ). 2

The approximation we have used becomes insufficient in the range x  xm

around the value x  xm 

 v0   ,    v1  

v02 . To find N  x  within this interval, where the density reaches its v1

maximum value, it is necessary to carry out calculation to a higher order over the small parameters being used, or to turn to numerical calculation. The spatial size of the appearing density bump can be estimated. As we see from (14), the density N  x  becomes of the order N 0 if x  xm fixed x

xm  v02 / v1 is equal to t

v02 / v1 . The corresponding time interval at a

v0 / v1 . This is in accordance with our initial estimate of d ,

see above. It is easily seen that the similar density peaks near x

xm  v02 / v1 (10) arise not only at

t  xm / v0 but also at time moments t  xm / v0  2n /  , where n is an integer.

6. NUMERICAL SOLUTION Introduce the dimensionless values: s   /  ;v1 /   v1;v0 /   v0 ;

 v x  p    t   ; X  2 x; q  1  Xv1  1  21 x. v0 v0  v0  Then

    x sX  sin    t   sin p        1  s / v0     v0 1   / v0    and the distribution function (8) may be presented in the form 2  s 1  q  / v1        f ( x, t , v)  F  s, p, q; v0 , v1   exp   s  v1 sin  p    . 1  s / v0         

N0

The electron density is obtained as a moment of the distribution function: N  x, t  









f ( x, t , v)dv 



f ( x, t , v ) d  



2   s 1  q  / v1        F  s, p; v0 , v1  ds   exp   s  v1 sin  p  1  s / v0  ds.        

N0



It is for this value (normalized to N0 /  ) that we present the results of numerical calculations. For a situation where the thermal velocity spread is much smaller than the directed velocity, v0 /   v0  10 , and the amplitude of the periodic modulation of velocity is significantly greater than this spread, v1 /   v1  3 , we show the shape of a surface that represents the dependence of density N on p and q , in Fig. 1. When v0 /   v0  30 and v1 /   v1  10 the “caustic” effect becomes even more pronounced than in the previous case. This is evident from a comparison of the two plots shown in Fig. 2. Also for this case, a number of plots is shown in Fig. 3 and 4, which shows how the time evolution of the density varies at a fixed distance x , if we change this distance. As it follows from the above analysis, see Sec. 5, the maximum “caustic” effect should be observed under condition t  x / v0 , i.e. at p  0 and q  1  v1 x / v02  0 . The presented plots

confirm this. We can see also the periodic recurrence of the effect for a given x occurring with the circular frequency  . 7. STRUCTURE OF ELECTRON PRECIPITATIONS Keeping in mind Eq. (6), we find that at a given characteristic distance d from the electron acceleration region to the level of their precipitation, density bunches can form described in the one-dimensional model in Sec. 5 and Sec. 6. Now let us turn to the three-dimensional structure. Following the simplest conjecture about relation between the spectral width and the correlation scale, we propose that the speed variation v (Eq. 5) in a particular field tube varies over time on the correlation scale of the order of the inverse wave frequency,

2 /  ; while on adjacent field

tubes it varies on the spatial correlation scale of the order of the transverse wavelength, Therefore, the position of a bunch D

2 / k .

v02 / v1 (see Eq. (6) and Eq. (10)) strongly varies both in

time and space, so that at the ionospheric level the bunches appear randomly, being spaced apart in the horizontal plane at the indicated spatial correlation scale, and oscillating in time on the indicated time correlation scale. 8. DISCUSSION AND CONCLUSION In this work we present feasible physical causes of a small-scale transverse structure in the flows of auroral electrons, generating a corresponding small-scale structure of discrete auroras. Based on results of a number of studies carried out in previous years, we believe that emergence of the three-dimensional current system involved in a disturbance of the type of magnetospheric substorm, produces intense field-aligned currents directed upward, from the ionosphere, which can be carried only by flows of electrons from the magnetosphere accelerated by a parallel electric field in the lower and middle part of the auroral magnetosphere. The parallel field leads to widening of the loss cone for these electrons, thus providing intense precipitating flows. However this field cannot penetrate deep into the ionosphere. An electrostatic double layer where quasi-neutrality is violated on a short vertical scale, limits that penetration. In this particular way two large scale (on

the order of hundreds of kilometers in the direction transverse to the magnetic field) features of the auroral plasma are formed. These are (1) “inverted V” structures in the auroral electron flows; (2) auroral cavity in the cold plasma of the upper ionosphere. Current-driven instability which arises in the auroral cavity occupied by the auroral electron flows, generates inertial Alfvén waves with a small cross-field scale of the order of one kilometer or less. Reaching large amplitudes, these waves undergo nonlinear interactions and form the shortwave Alfvénic turbulence. An important feature of the inertial Alfvén waves is a nonzero parallel electric field. This field has backward influence on the auroral electron flows, leading to their acceleration/deceleration. In every flux tube, in the field of many wave packets, this provides a stochastic effect of the flow velocity modulation, occurring at characteristic timescales of the turbulent waves. In the direction transverse to the magnetic field, the spatial scales of the effect are determined by the transverse wavenumber spectrum of the wave turbulence. These processes take place high enough in the lower and middle part of the auroral magnetosphere, at altitudes  104 km. Therefore, it may be considered that at those altitudes the flow velocity modulation occurs, and at lower heights the electrons form a freely moving flow, down to the ionospheric level,

100 km. Then a nonlinear effect should take place in the flow,

relating to kinetics of freely moving particles: fast particles are catching up slower ones, and electron density intensifications are forming. An important point here is that in the region of auroral upper ionosphere – lower magnetosphere, the energetic auroral electrons are really an external beam penetrating the background plasma from above. Thus a specific kinetic feature arises, and in this situation the plasma evolution would not follow any fluid scenario proposed in some simulations (e.g. [Chaston and Seki, 2010]). The electron acceleration region, ranging up to a few RE , is located generally much higher than the region of interaction of accelerated electrons with the atmosphere, at the

altitude of auroras, about one hundred kilometers, and therefore much higher than the region where a strong horizontal structuring of electron flows resulting in the fine structure of discrete auroras, is observed. The scenario proposed in this paper involves the following items. (1) The strong fieldaligned current carried by electrons accelerated up to energies of

1  10 keV, leads to current-

driven instability of Alfvén waves which, within the auroral cavity, are inertial Alfvén waves with small transverse scales  1 km. (2) The resulting wave turbulence, while interacting with fast electrons, generates variations of their speed at the characteristic frequencies of the turbulence, and at corresponding horizontal spatial scales. (3) The relatively weak temporal variation of the flow velocity occurring at high altitude, in the region of electron acceleration, results at the low altitude of auroral precipitation, in a strong variation of the flow density. This is due to a well-known nonlinear effect in the kinetics of freely moving particles: fast particles catch up those moving slower, and electron density bunches are formed. (This effect is completely similar to the effect of caustic in the ray optics.) (4) Strong transverse, horizontal inhomogeneities of electron flow density give rise to corresponding inhomogeneities of auroral luminosity, i.e. to small-scale structure of discrete auroras. These items are presented in Figure 5, a cartoon showing our key points. The Figure includes also a schematic of formation of bunches in a klystron. The approach proposed in this paper, provides a likely reply to the following key question. Finely structured active auroras occur inside much wider “inverted V” electron precipitations; all the electrons forming that wide “inverted V” have gained their energy high over the precipitation altitude. Then how can the luminosity effect vary so strongly on a short spatial scale being produced by an electron beam almost uniform on such a scale? It must be that the initially almost uniform flow decays into small-scale bunches at the ionospheric heights. So the proposed here nonlinear “caustic” effect looks the most feasible explanation.

We have used the known observational data relating to the intensity and spatial scales of the Alfvénic turbulence, and the flow velocity modulation parameters have been evaluated. The kinetic theory and numerical calculations show how this modulation may really give rise to appearance of strong inhomogeneities in the density flux at the ionospheric level. While considering the resulting connection of this effect with the Alfvénic turbulence, it is necessary to take into account its dependence on characteristic variations of the electron beam velocity which have correlation scales in time and space determined by corresponding scales of the Alfvénic turbulence. This provides an explanation of the well-known similarity of horizontal spatial structure of the Alfvénic turbulence and that of discrete auroras. Note that a significant contrast in discrete auroral forms, i.e. the difference in the intensity of highs and lows, corresponding to the intensity index differing, say, by one unit, occurs when the energy flux differs nearly by one order of magnitude. Numerical calculations have shown that such a distinction is quite feasible in our scenario of the auroral flow structuring. However, as mentioned in Sec. 5, considering the flow of accelerated electrons free and taking into account the nonlinear “caustic” effect only, we actually oversimplify the situation. Indeed the formation of electron bunches must be accompanied by formation of an uncompensated electric charge. It generates an electrostatic field repelling electrons of the bunch. I.e. the electron motion is no more free, it is necessary to take into account the force exerted by that electrostatic field. Bearing in mind relatively low frequency disturbances, small scale in the direction transverse to the magnetic field, we must take into account also the motion of ions allowing to compensate for the excess charge. Thus we come to that same situation as described in the twofluid plasma model, which in the linear approximation provides inertial Alfvén waves (see Sec. 2). However, at low altitudes where the nonlinear “caustic” mechanism is effective, the disturbance, as we can see, seems to become a nonlinear wave.

In fact, this problem may be not so significant. Calculations of Sec. 5 show that density bunches have a small vertical scale, and they can be formed at low heights. And if such a bunch is completely below the electrical double layer separating the dense ionospheric plasma from the auroral cavity, then its density is very low as compared to the background one, so that actually there appears no problem of its charge compensation by ionospheric ions and electrons. Specific comprehensive calculation of the nonlinear perturbation amplitude, which should take into consideration the vertical non-uniformity of the problem (with magnetic field mirroring involved), is beyond the scope of this work. Here we only pointed out the importance of the effect and gave its preliminary evaluation. Note also that our approach is based on the properties of the wave turbulence obtained in the linear approximation for the inertial Alfvén waves, which, generally speaking, is incorrect from another point of view. The analysis carried out in [Golovchanskaya et al., 2011] showed that the Alfvénic turbulence in the upper ionosphere/lower magnetosphere cannot be interpreted purely in terms of inertial Alfvén waves (although the presence in them of the wave component with power of ~ 10% of the total signal power is quite likely). It was found out that the main properties of the Alfvénic turbulence (a number of its statistical features, its chaotic polarization fully different from the regular time dependence of the transverse magnetic field vector expected for a plane wave, and others) may be explained as a result of nonlinear interaction of coherent Alfvénic structures [Pokhotelov et al., 2003; Chang et al., 2004] with a magnetostatic component being present. This, however, again relates to the heights generally lower than those which are meant in this paper for the region where the electron flow speed variations are generated. And anyway this does not deny the action of stochastic mechanism of auroral electron flow modulation, which we point out here, though the issue may have a significant impact on the quantitative estimates obtained.

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N

q

p

Figure 1

N N 16

14

12

10

8

6

4

2

2.5

0

2.5

5

7.5

p Figure 2 v0 /   v0  10 ; v1 /   v1  3 - thick line v0 /   v0  30 ; v1 /   v1  10 - thin line

10

12.5

N N 16 14 12 10 8 6 4 2

2.5

0

2.5

5

7.5

p Figure 3 X=0.1

- thick solid line

X=0.05

- thin solid line

X=0.02

- thick dashed line

X=0.005 – thin dashed line

10

12.5

N N

15

12.5

10

7.5

5

2.5

2.5

0

2.5

5

p Figure 4 X=0.1

- thick solid line

X=0.13 - - thin solid line X=0.18 - thick dashed line X=0.25 – thin dashed line

7.5

10

12.5

Figure 5

FIGURE CAPTIONS Figure 1. Dependence of the number density N on the dimensionless parameters p and q . Figure 2. The “caustic” effect for different parameters of the initial distribution function. Figure 3. Time dependence of the number density at a fixed distance x , at variation of that distance. Figure 4. Time dependence of the number density at a fixed distance x , at variation of that distance (continued). Figure 5. Formation of the small-scale structure of auroral electron precipitations under strong upward field-aligned current conditions (schematic).

Highlights 

Alfvénic turbulence modulates energetic electron flows in the auroral magnetosphere

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At ionospheric level this leads to formation of strong electron density peaks

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Emerging electron flow bunches lead to small-scale structure of discrete aurora