ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-999-2017Electron cyclotron maser instability (ECMI) in strong magnetic guide field reconnectionTreumannRudolf A.https://orcid.org/0000-0002-9783-994XBaumjohannWolfgangwolfgang.baumjohann@oeaw.ac.athttps://orcid.org/0000-0001-6271-0110Department of Geophysics and Environmental Sciences, Munich University, Munich, GermanySpace Research Institute, Austrian Academy of Sciences, Graz, Austriavisiting scientist at: International Space Science Institute ISSI, Bern, SwitzerlandWolfgang Baumjohann (wolfgang.baumjohann@oeaw.ac.at)28August2017354999101314October201620July201726July2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/35/999/2017/angeo-35-999-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/999/2017/angeo-35-999-2017.pdf
The ECMI model of electromagnetic radiation from electron holes is shown to
be applicable to spontaneous magnetic reconnection. We apply it to
reconnection in strong current-aligned magnetic guide fields. Such guide
fields participate only passively in reconnection, which occurs in the
antiparallel components to both sides of the guide-field-aligned current
sheets with current carried by kinetic Alfvén waves. Reconnection generates
long (the order of hundreds of electron inertial scales) electron exhaust
regions at the reconnection site X point, which are extended perpendicular to
the current and the guide fields. Exhausts contain a strongly density-depleted hot electron component and have properties similar to electron
holes. Exhaust electron momentum space distributions are highly deformed,
exhibiting steep gradients transverse to both the reconnecting and guide
fields. Such properties suggest application of the ECMI mechanism with the
fundamental ECMI X-mode emission beneath the nonrelativistic guide field
cyclotron frequency in localized source regions. An outline of the mechanism
and its prospects is given. Potential applications are the kilometric radiation (AKR) in auroral
physics, solar radio emissions during flares, planetary emissions and
astrophysical scenarios (radiation from stars and compact objects) involving
the presence of strong magnetic fields and field-aligned currents. Drift of
the exhausts along the guide field maps the local field and plasma
properties. Escape of radiation from the exhaust and radiation source region
still poses a problem. The mechanism can be studied in 2-D particle
simulations of strong guide field reconnection which favours 2-D, mapping the
deformation of the electron distribution perpendicular to the guide field,
and using it in the numerical calculation of the ECMI growth rate. The
mechanism suggests also that reconnection in general may become a source of
the ECMI with or without guide fields. This is of particular interest in
extended turbulent plasmas where reconnection serves as an integral
dissipation mechanism of turbulent energy in myriads of small-scale current
filaments.
Radio science (waves in plasma) – solar physicsastrophysicsand astronomy (radio emissions)Introduction
The present paper is intended to present a scenario, not a fully developed
theory, for the generation of intense radio radiation in a strongly
magnetized plasma by the electron cyclotron maser instability (ECMI)
mechanism. This mechanism is a non-thermal source of radiation in radio waves
which we do not review here in detail see, e.g.,for a still not
outdated review. In the recent past it has experienced a few
particular but no fundamental specifications for recent examples
one may consult respectively complications (inclusion
of electron beams, calculation of low harmonics, numerically obtained graphs
of growth rates, etc.). The new scenario contrasts these attempts in so far as
it opens up a new direction in ECMI research. We do not do any calculation
because the problem is too complex. A quantitative treatment requires a
full-particle simulation including radiation. This will become clear when
presenting the idea and describe the mechanism. The idea and scenario are
interesting enough for injection and stimulation of a quantitative treatment
by researchers working in the two sectors: collisionless reconnection (by
numerical particle-in-cell (PIC) simulations) and generation of electromagnetic radiation (using
the properties inferred from the PIC simulations in a numerical calculation
of the ECMI growth rates).
Review and justification
In order to provide a brief justification for attempting a new twist in ECMI
theory we just mention that generation of high-intensity free space waves in
a plasma, though for prospective reasons highly desirable from the
geophysical, space and astrophysical viewpoint, poses a major problem. Any
escaping radiation is based on the dynamics of electrons. In the electron
equation of motion it is bound to the third time derivative
⋯v of the electron velocity
v. Gyro and synchrotron radiation is thus weak (providing very weak
energy losses only), though it is comparably easy to treat see,
e.g., while, when detected, it maps temperature, magnetic field
strength and mean plasma density see, e.g.,. Guided by
the whistler instability, nonthermal radiation in the free space modes X and
O from hot plasmas has early on provoked attempts to base it on thermal
anisotropies in the electron distribution
. These
anisotropies, however, turned out to be unreasonably large for causing instability,
provoking doubt in this mechanism see, e.g., compared
to synchrotron radiation. The breakthrough see, following
the earlier suggestions , was
brought by the observation that it is the relativistic effect in the electron
distribution, if properly accounted for in the resonance curve, which must be
added in order to obtain positive growth rates and generate high radiation
intensities. Positive perpendicular-momentum space gradients in the
weakly relativistic electron distribution turned out to be the main
necessary condition for instability. Radiation in the fundamental
X mode was found beneath but close to the local (non-relativistic) electron
gyrofrequency ωc=eB/m. Its excitation requires, in addition, as a
sufficient condition, that the plasma density is low, specifically
the ratio of plasma-to-gyrofrequency (ωe/ωc)2≪1 must be
sufficiently low for reaching substantial growth rates and, hence, radiation
intensities.
The simple physics behind the ECMI is that, at such low densities, plasmas
are unable to absorb the unstably generated waves. There is not enough plasma
to digest the extra free energy. As a matter of fact, the local absorption
coefficient turns negative, and the whole plasma region becomes an emitter of
radiation, mainly in the X mode and weaker in the O mode. There is, of
course, also excitation of the Z mode below the upper hybrid frequency
see, e.g.,, but this is basically electrostatic and cannot
leave the plasma without invoking other wave transformation processes.
Modes, harmonics, thermal levels
The most general expression of the ECMI growth rate for the X mode can be
found in for arbitrary weakly relativistic electron
distributions see alsofor application to
AKR. In principle it gives the growth rate
ℑ(ωℓ) of the ECMI for any electron cyclotron harmonic
ℓ=1,2,3… of ωℓ∼ℓωc. Inspection shows that,
to leading order in ℓ, the growth rate decreases with order as
ℑ(ωℓ)∼ℓ-1. Observations of the X mode, mostly in the
auroral region, do indeed indicate the presence of low harmonics as
was noted byand others.
Any wave must
grow from its thermal level. In view of the fact that the thermal level of
the fundamental X mode is very low (see Fig. ) and given its
restricted possibilities of escape, it becomes questionable whether the
observed AKR is indeed propagating on the fundamental. It is probable
that it propagates on one of the lower cyclotron harmonics, because many
observations locate its frequency at 300–700 kHz, which for the fundamental
poses the source rather deep into the upper
auroral ionosphere.
In fact, for those harmonics, if excited sufficiently
strongly, the above necessary condition is obsolete as higher harmonics can
propagate out and can be observed remotely. Note that AKR
radiation observed remotely is possibly not in the fundamental, but
harmonic, putting its source a bit higher in the upper auroral ionosphere.
Excitation of any wave by instability requires the presence of a (weak)
thermal background fluctuation at the unstable frequency from which it can
grow for the kinetic theory of electromagnetic fluctuations see,
e.g.,.
Figure after shows that the
electromagnetic thermal fluctuation spectrum (for almost perpendicular
propagation) in a thermal magnetized plasma is structured harmonically by
electromagnetic Bernstein modes. These fluctuations intensify at long
wavelengths (small wavenumbers k) along the propagating X-mode
dispersion (for the parameters chosen in this case starting at the fifth
harmonic). Hence, since the ECMI unstably excites higher harmonics in the X mode, these encounter an appropriate thermal level from which they can grow.
Note, however, that the thermal level for the fundamental (ℓ=5 in this
case) is rather weak compared to the next low harmonics, implying that low
harmonics will initially grow faster than the fundamental. On the other hand,
if the ECMI were able to also excite electromagnetic Bernstein modes,
then these modes would find plenty of thermal fluctuations available which
they could amplify to become free-space harmonically structured propagating
modes at shorter than X-mode wavelengths. A calculation like this one is
still outstanding but might be worthwhile to perform in view of the broad
prospects of applications (for instance in explanation of harmonic emissions
during solar radio events, like in type I and type IV radio bursts, fibre
bursts, and possibly even radio emission from Jupiter).
The most interesting mode for the purposes of the following model, which does
not exclude the possible excitation of any of the higher cyclotron harmonics,
is the (fundamental) X mode. The fundamental cannot leave the plasma, because
if driven unstable by the ECMI it propagates at frequency
ω≲ωcbelow its high-frequency cutoff on the
lower (trapped) X-mode branch. Thus, escape of the fundamental mode to become
visible requires propagation out of the source region to lower magnetic field
strengths. This is particulary true for the most famous region of excitation
of the ECMI, the auroral upper ionosphere/lower magnetosphere of Earth (where
it is the typical case). In order to have the wanted efficiency of the
mechanism, complicated if not exotic forms of the electron distribution need
to be imposed. Among those a variety has been investigated in the literature,
with global loss-cone distributions see, e.g.,for elaborate
examples of steep perpendicular momentum space gradients
traditionally having been favoured.
The thermal spectral densities of electromagnetic waves in a
magnetized plasma for nearly perpendicular propagation
after for an electron plasma-to-cyclotron frequency ratio
of ωe/ωc=5. The thermal level is structured by the
electromagnetic electron cyclotron harmonics (Bernstein modes). The red
intensified spots in the thermal noise starting at ℓ=5 are located along
a parabola of rising frequency, which is the free-space branch of the X mode
in combination with the electromagnetic Bernstein modes. They may serve as
cyclotron harmonic seed electromagnetic fluctuations for the X mode, which,
when becoming amplified by the ECMI, can propagate and escape from the
plasma.
Localized sources: electron hole model
In some more recent papers we proposed that
electron holes, generated by strong field-aligned currents in a
magnetized plasma, could possibly become sources of the ECMI. The existence
of electron holes in both the downward current region see,
e.g., and in the upward current region
has been observationally confirmed,
with the abundance of such holes in the downward current region being
overwhelmingly higher. This electron hole model of the ECMI has the
(technical) advantage that the relativistic resonance curve in momentum space
becomes a circular section, a fact that substantially simplifies the
calculation. In addition, it reproduces the striking narrowness of the
emissions in frequency space. Moreover, the Doppler shift of the emissions
caused by the motion and dynamics of the electron holes nicely reproduces the
observed modulations in the narrow electron cyclotron emission lines.
Excitation in the downward current region, where the strongest auroral-zone
field-aligned currents flow and electron holes are ubiquitous, directly
explains the observation of strong radiation in the upward current region by
the simplicity by which the radiation can escape from the downward to the
upward current region across the steep transverse plasma gradient between
adjacent two of them.
However, the great and so far unsurmountable disadvantage of that
model is that electron holes are Debye-length objects on scales
λD=ve/ωe (with ve the electron thermal speed). Such
spatial structures should barely be able to excite strong radiation at
wavelengths of several kilometres as they are simply too small. One would need to deal
with a spatial distribution of very many such holes located in a volume over
one radiation wavelength in order to determine their collective (overlap of a
series of holes by a much longer unstable wavelengths) and statistical
(contribution of very many holes distributed in the volume to wave
excitation) effects which might smear out most of the fine structure. Such a
calculation, though certainly worth being applied in the electron hole
radiation mechanism, has not been performed to date.
Requirements on an efficient localized source
In order to cure the deficiency of the short scale, it becomes necessary to
look for other desirably common and possibly more frequent reasons for the following:
the generation of electron depletions of sufficiently large spatial extension
L≫λD;
that these regions should contain sufficiently anisotropic or otherwise strongly deformed, just weakly relativistic electron momentum
distributions;
in sufficiently strong magnetic fields, thereby maintaining the advantages of the electron hole model.
Collisionless reconnection: relevance for the ECMI
A most promising candidate mechanism of this kind is collisionless
reconnection, which is well known to locally generate regions of vastly
(sometimes in the simulations up to ≳90 %) depleted electron
densities, containing strong electric potential drops which cause electron
heating and acceleration, and generate violently deformed
electron-momentum-space distribution functions.
Reconnection has so far not been considered in relation to the ECMI.
However, a large number of kinetic-numerical PIC (particle-in-cell)
simulations of collisionless reconnection have in the past 3 decades been
put forward in two (2-D) and recently also in three (3-D) dimensions.
Therefore, the relevant physics (still excluding many details) of
collisionless reconnection under various different settings is, by now, quite
well understood, mainly under conditions when two magnetic fields of equal
(or not vastly different) strengths and mutually moderately inclined
directions collide, merge, and annihilate their antiparallel components, i.e.
undergo reconnection for recent reviews of related aspects of
collisionless reconnection see, e.g.,. In this
process the energy of the annihilated fields goes into the electric potential
which causes the mentioned electron depletions, heating, and acceleration.
(For more general aspects of reconnection, including collisions and
laboratory effects, the reader is referred to , and
.)
In the following, when talking about reconnection, we mean
collisionless reconnection as this is the only form of reconnection
which is relevant on the scales under consideration. From the point of view
of the physics of hot diluted plasmas it is highly questionable whether any
of the originally proposed collisional (magnetohydrodynamic) versions of
reconnection is realized anywhere else in nature other than in the deep
molten metallic interiors of planets, where resistive reconnection
participates in the magnetic dynamo action, and the highly compressed
resistive stellar convection zones above the cores, where it participates as
an important energy dissipation process in nuclear fusion.
Weak guide (Bg=0.5B0) field PIC simulations of collisionless
spontaneous reconnection in 2-D after with
current and guide field perpendicular to the plane. From top to bottom:
normalized density, normalized magnetic field, Hall field structure, electric
reconnection potential. The density exhibits the asymmetric exhaust (deep
blue) and (red) electron accumulation regions caused by the E×Bg
drift in the reconnection electric and guide fields. The nearly rectangular
box reconnection region centred at the X point is characterized by a weak
magnetic field. Very weak Hall fields arise due to the immobile ion
background which for our purposes are of no interest. They just contribute to
asymmetry. The electric potential shows region of different signs equivalent
to the density structure. Note that in this writing positive potentials
attract, negative potentials expel electrons. Note the very long horizontal
extension of the reconnection site which reaches a length of roughly L∼300λe which means ±150λe to both sides in spite of the
fact that the electrons in the guide field remain partially magnetized. The
ion inertial length λi≈43λe implies that the
reconnection site is elongated on each side by up to ≳3λi.
The magnetization of the electrons in the guide field perpendicular to the
reconnection field affects the reconnection process only very little.
Reconnection as generator of electron quasi-holes
So far reconnection has been investigated mainly in counterstreaming
flows/colliding magnetic fields under small angles and for
plasma β=2μ0NTe/B02>1. Under such conditions the magnetic field
is weak and, though diluted, the surrounding ambient plasma remains
superdense: ωe>ωc. Clearly the ECMI, even if excited in the
fundamental X mode at the reconnection site, will (a) be of very weak
intensity, because the dissipated magnetic energy is comparably small, and
(b) will – similarly to the electron hole model – be trapped in the local
density depletions near the X point, thus unable to escape. This case is of
no interest here though, as noted above, at higher harmonics the underdensity
condition is of lesser importance. One may thus expect that reconnection
sites, even under weak field conditions, may show some radiative signatures
in the low-electron-cyclotron harmonics thereby mapping the magnetic field
strength around the reconnection sites.
Recognizing this possibility
is of substantial interest in astrophysics as a mechanism that may cause a
large volume of highly turbulent plasma to become glowing in radio emission
from the myriads of small-scale reconnecting current filaments undergoing
spontaneous reconnection . Such a radio glow is
conventionally interpreted as synchrotron radiation but may indicate something
completely different: turbulence, magnetic field strength, plasma density
obtained from the electron inertial scale, reconnection and dissipation of
turbulent energy in the many small-scale current filaments. Though the
dissipation mechanism has not yet been investigated in detail, its
clarification would in this case provide important information about the
matter involved.
An example of a 2-D weak guide field simulation of reconnection is
depicted in Fig. for the case of a guide field
Bg=12B0, half the value of the reconnection field B0. The
electron density (first panel) is seen to be asymmetric with respect to the
central X point where the magnetic fields ±B0 from above and below come
into contact and reconnect, with magnetic field strength strongly reduced
(second panel). The density near the reconnection site in the current centre
exhibits two regions of density increase (red) and two regions (dark blue)
where the electrons are depleted, indicated as exhausts. These
exhausts strongly resemble the main property of electron holes which are a
depletion of the electron density on the Debye scale. Electron exhausts thus represent substantially more extended electron quasi-holes.
Exhaust properties of relevance for the ECMI 1
The exhaust length is of the order of ≳100λe or more, with
λe=c/ωe the electron inertial length, and width the order of
20λe. The ratio of electron inertial to Debye length is
λe/λD=c/ve≳10≫1, making the size of the exhausts
substantially larger, by roughly a factor of several 100, than the size of an
electron hole in both directions perpendicular to the ambient (guide) field.
A large size like this one is clearly advantageous in view of the ECMI and
its wavelength. In fact, since λe∼1 km or so in the auroral
ionosphere, the size of the exhaust allows for wavelengths of several tens of kilometres
to propagate inside the exhaust (as weak thermal fluctuations ready for
amplification). This makes an exhaust a valid source for AKR, if only the
conditions for excitation can be satisfied.
Since the auroral ionosphere is the canonical paradigm of an ECMI source,
such conditions are in favour of the assumption that whenever reconnection in
the auroral ionosphere can proceed, the exhaust region will serve as source
of AKR as the result and signature of the ECMI.
The release from the noted deficiency in magnetic field strength – and thus
in cyclotron frequency – is to assume that reconnection takes place in a
so-called magnetic guide field Bg, which points
along the current flow J and is of sufficient strength
Bg≫B0 to by far exceed the proper magnetic field
∇×B0=μ0J of the guide-field-aligned current.
This, however, is exactly the configuration encountered in regions where the
ECMI is believed to be at work, for example in Earth's auroral upper
ionosphere, where the ambient geomagnetic field is a factor of > 100
stronger than the magnetic field B0 of the field-aligned current.
Similarly, conditions in the strong magnetic fields in Jupiter's or Saturn's
lower magnetospheres are also promising.
The example given in Fig. is typical for weak or moderate
guide field simulations of reconnection. ECMI source regions of interest here
are, however, characterized by strong magnetic guide fields. Reconnection
simulations in such fields are not available yet. They have not attracted any
interest in the reconnection community, because reconnection is considered a
mechanism for the release of magnetic energy and particle acceleration rather
than radiation. Radiation, as mentioned above, is just a minor fraction of
the released energy. The density of the reconnection-released energy is, at
maximum, the total energy of the reconnecting magnetic fields ≲2|B0|2/μ0. Because strong magnetic guide fields Bg along the
current J are not affected by reconnection, this maximum is
independent of the guide field strength. Concerning reconnection, the guide
field effect is twofold: strong guide fields maintain the reconnection 2-D, and in addition they magnetize the electrons (as also the ions) in the plane
perpendicular to the guide field.
Remarks
At this place there is an important point to make. Guide field simulations of
reconnection in 2-D were occasionally performed with guide field strengths
in the range 0.1≤Bg/B0<2. Though 2-D simulations with very weak guide
fields are not completely unreasonable however, see the 3-D
simulations reviewed by, two-dimensionality breaks down
when Bg/B0≳1/2, because then the total magnetic field becomes a
spiral of opening (winding) angle α∼tan-1(Bg/B0). Results of
2-D simulations become unreliable in this case, as the field is
3-D. However, at dominating guide fields Bg≳5B0, say, when the total magnetic field stretches, the stiff guide field
restores two-dimensionality, which becomes robust then. This is clearly the
case for reconnection in the auroral upper ionosphere with its strong
geomagnetic field and the comparably much weaker fields of any field-aligned
currents.
Similar arguments apply of course also to Jupiter and other strongly
magnetized planets , the Sun, and stars. For this reason, reconnection has
never been seriously considered to take place and become important here in
the ECMI. Instead the generation of anomalous resistivities by
plasma instabilities has been invoked, and even the envisaging of something called “breaking of field
lines” which naturally would violate the basic laws of electrodynamics.
The proposed model
Having set the framework, we are ready to turn to the
physical model settings. This model consists of the following three parts:
(a) the reconnection in strong guide fields, which is the basic condition for
(b) the mechanism of ECMI, which generates the radiation field, and
(c) the conditions for propagation and escape from the plasma.
Requirements for reconnection
Reconnection is based on the assumption of annihilation of oppositely
directed magnetic fields, which belong to a sheet current of comparably
narrow width. Such narrow current sheets exist in the auroral magnetosphere
with current flowing along the strong geomagnetic field downward into the
ionosphere where they close with perpendicular currents and back upward along
the field in some slightly broader sheets. These currents are carried by
electrons which flow upward respectively downward in the adjacent
field-aligned regions.
Magnetic field-aligned currents of this kind are pulses of kinetic
respectively inertial Alfvén waves (KAW, IAW), depending on whether the
plasma β>m/mi or β<m/mi, i.e. larger or smaller than the
electron-to-ion mass ratio. At high altitudes in the comparably weak
geomagnetic field and high plasma temperatures these are KAW having transverse
dimensions of the ion gyroradius rgi, corresponding to broader current
sheets. When entering the auroral source region of AKR they become IAW of
transverse scale of the order of the electron-inertial length. This behaviour
can be read from the KAW dispersion relation see, e.g.,Eq. 10.181,
revised edition 2012, which in the upper ionospheric IAW
region becomes
ωkA2≈k‖2VA21+k⟂2λe2,k‖2≪k⟂2∼λe-2.
It propagates along the guide field at a fraction of the Alfvén speed while
it belongs to a narrow current sheet of typical scale of the order of
λe. Thus the upper ionospheric field-aligned current system is
structured, consisting of a series of electron inertial-scale current sheets
varying with time. Each sheet carries its proper magnetic field B0.
This field is perpendicular to the main geomagnetic guide field
Bg⟂B0 and has opposite direction to both sides of the
current sheet. B0≪Bg is a weak field, substantially weaker
than the (geo)magnetic guide field along that the current flows. Such a
current total magnetic field (J,Bg+B0) configuration is
schematically shown in Fig. .
Two oppositely directed electron-scale current sheets with current
J‖ (yellow) flowing along the guide magnetic field in the case that
Bg>B0 is stronger than the magnetic field B≡B0 (light-blue
vectors) of the guide-field-aligned current. The currents close via J⟂
somewhere in a dissipative layer (red) where they are allowed to flow
perpendicular to the field. The width of the guide-field-aligned current
sheets is assumed to be narrow, the order of the electron inertial length
which would allow the current magnetic field B0 to reconnect across the
current sheet.
Once the current sheet becomes narrow, of the width of either the electron
gyroradius or electron inertial scale, the electron magnetization decreases
due to partial electron demagnetization until it can no longer balance the
attractive Lorentz force between the two antiparallel magnetic fields on both
sides of the current sheet. Once this happens, the two oppositely directed
fields collapse, and spontaneous reconnection onsets.
There is a slight complication in this process, because the electrons are
still magnetized in the guide field. However, this magnetization is in the
plane perpendicular to the guide field. It causes a weak orbital diamagnetism
along the guide field which slightly reduces the guide field but has no
effect on the attractive Lorentz force. Hence, under these conditions the
magnetic field of the field-aligned current carried by the IAW will collapse
locally and reconnect. Locally its two oppositely directed components are
perpendicular to the guide field and in the plane perpendicular to the
current across the guide field in the centre of the current sheet until they
mutually annihilate, which causes an X point in the magnetic field of the
current and the IAW to which the current belongs. The IAW develops a vertex
in its magnetic field B0, the magnetic X point, at this location
while the field-aligned current continues flowing. The IAW also continues
propagating along the field, which causes a shift of the whole reconnection
region, the X point, along the field either to higher or to lower field
strengths, depending on the direction of the IAW-wave propagation. The guide
field remains unaffected by this whole process of reconnection.
Role of IAW
In order to avoid confusion it becomes necessary to clarify the role of the
KAW. This wave is required as the wave-carrier of the current. It possesses a
guide-field-aligned electric field E‖kA=-ik‖ϕ‖kA which
accelerates electrons along the guide field to become the field-aligned
current
Je‖kA=-m2μ0TeωkAk‖λe2ϕ‖kA=im2μ0TeωkAk‖2λe2E‖kA
and it is the KAW magnetic field, the magnetic field
B0⟂Bg of the KAW current, which is perpendicular
to the strong guide field Bg. In the downward current region the
potential ϕ‖kA>0 is positive with upward-pointing gradient, and
both current and electric field point downward, accelerating electrons
upward. In the adjacent upward current region the directions are opposite.
Accordingly, the current-carrying kinetic Alfvén waves propagate downward
respectively upward in these regions.
The KAW magnetic field B0 reconnects at some location where the
transverse pressure of the magnetized electrons in the wave vanishes on the
electron inertial scale <λe and thus can no longer balance the
attracting Lorentz forces between the antiparallel fields ±B0 to
both sides of the field-aligned current layer (the general reason for
spontaneous onset of collisionless reconnection). In Earth's upper auroral
ionosphere this happens within a few Earth radii in the region where the KAW
changes from its kinetic to its inertial limit with the inertial Alfvén
wave (IAW) dominated by electron inertial effects. This happens at a
geocentric radial location where in the strong geomagnetic guide field the
plasma-to-magnetic energy density ratio β=2μ0NTe/Bg2 changes from
β>m/mi to β<m/miseeSect. 10.6.5.
Thus, for reconnection to occur in a strong guide field set-up like in the
case of the auroral upper ionosphere, the parallel electric field of the IAW
and the related electron acceleration along the (ambient geomagnetic) guide
field is important as the provider of the current source. Without its
presence nothing would happen, i.e. without the IAW and its parallel electric
field, whose sources must be seen at a different place, there would be no
field-aligned current and thus no reconnection. Otherwise the IAW's
guide-field-parallel electric field has no effect on the reconnection itself.
Ultimately, though, after reconnection starts and evolves to a violent state,
it is the magnetic energy of the IAW which in the reconnection process at the
reconnection site is transformed into heat, entropy, particle acceleration
and radiation, not the energy of the main guide field. Thus, for the IAW,
reconnection is a local dissipative process.
This is a
sensitive point. Dissipation of the wave-magnetic field implies that the IAW
at the reconnection X point breaks off. Thus, when reconnecting, the IAW
decays into a wave chain. Because the parallel momentum is unaffected, both the
wave chain and exhausts should continue moving along the guide field. To
simulate this would be most interesting, because of the expected effects in
the aurora.
It affects or even destroys the IAW locally over the volume of
the exhaust while leaving the guide field unaffected – at least as long as
one deals with weakly relativistic plasmas. In strongly relativistic plasma
the situation should become more violent
because shock waves and turbulence may form.
So far we have mentioned only the parallel IAW electric field and its principal
effect. IAWs possess substantially stronger than parallel perpendicular
electric fields E⟂kA. These are transverse, thus being
perpendicular to both the wave and ambient (guide) magnetic fields. Since, as
explained above, electrons in a strong guide field are magnetized in the
plane perpendicular to the guide field, the effect of the perpendicular KAW
electric field is that it causes an E⟂kA×Bg-drift of the
exhaust electrons. Qualitatively, this drift is along the exhaust and thus
along the reconnecting field B0. It acts amplifying and modulating
the acceleration of the electrons along the exhaust. In this way it
contributes to the deformation of the electron momentum distribution in the
direction perpendicular to the ambient magnetic guide field. Its presence is
therefore quite important for the ECMI though of minor importance for the
reconnection. Also, it just adds a constant speed along the exhaust. To
quantitatively infer its contribution requires numerical PIC simulations. In
contrast, the weaker reconnection electric field which is along the exhaust
and parallel to B0 causes a continuous acceleration of the gyrating
electrons when experiencing a large number of cyclotron gyrations in the
exhaust, as has been noted earlier. This is an effect that is particulary
suited for investigation in simulations.
Reconnection occurs at the expense of the IAW magnetic field, i.e. the
wave-carried field-aligned current. Both suffer energy losses caused by
reconnection, but these are grossly independent of the guide field, in
particular in strong guide fields. It is only in weaker guide fields Bg∼B0 where the total field becomes wound up and the whole reconnection
process is 3-D that, therefore, the guide field energy contributes to
reconnection. In the auroral ionosphere we are confronted with the example of
a strong guide field case.
Historically, reconnection
has never been considered to be of any relevance for processes going
on in the auroral region with its strong ambient field. It has long been known
that KAWs and IAWs play an important role as current carriers,
accelerating electrons e.g. via their parallel wave
electric fields. KAW soliton formation has been considered as a nonlinear
effect see, e.g.,and others, and
current-driven ion and electron holes
see have been identified (mainly from
FAST and Viking spacecraft observations). However, the main dynamical effects
were attributed to field-aligned current-generated high anomalous
resistivities in the aurora see, e.g., and some kind of
“field-line breaking”. Anomalous resistances turned out to be rather inefficient,
while field-line breaking is physically questionable. Reconnection in strong
guide fields, if experimentally confirmed, cures most of these deficiencies,
though it still requires simulational and experimental verification. Locally
it generates a region of high wave-magnetic dissipation, particle
acceleration, and radiation signatures in ECMI. Statistically, the
dissipation could possibly even be described as a localized anomalous
wave-generated resistance. However, the basic mechanism is not to be sought
in any wave nonlinearity; rather it is the by now well-established and
violent reconnection process.
Effect of reconnection
In strong guide fields the effects shown in Fig. should
with high probability evolve in a much more pronounced way, causing deep
electron exhausts, large asymmetries and even stronger acceleration/heating
and anisotropies of the exhaust electron component.
Schematic of the electron dynamics inside the electron exhaust
(rectangular yellow region) in the presence of a strong guide field (black
crossed circles) directed into the plane and antiparallel to the current. The
exhaust is shown as a rectangular region with reconnected magnetic field lines
B. The blue arrows represent the reconnection electric field which points
into the centre of the X point. Without guide field this field would expel
and accelerate electrons out of the reconnection site, as is known from the
non-guide field simulations. With strong guide field, however, it first
forces the electrons to perform a E×Bg drift upward to the left and
downward to the right until the electrons accumulate at the exhaust boundary
where the field ceases to exist and they are reflected from the current
magnetic field B. This forms the green electron quasi-“holes” and the
dense red regions of accumulated electrons. Reflection at the exhaust
boundary forces the electrons to perform meandering motions along the guide
field/current direction. During each such half-gyration they on their turning
points experience the reconnection electric field and like in a synchrotron
become accelerated in the direction opposite to the electric field. This
causes a stepwise increasing gyroradius shown here in the projection onto the
plane perpendicular to the guide field. In effect these electrons ultimately
reach high energy in the direction of the exhaust perpendicular to Bg
causing deformation of the momentum space distribution. This is the condition
required for the ECMI to become excited.
Figure gives a schematic representation of the various
processes in the reconnection region with the strong guide field and electric
current pointing into the plane. According to the simulations with weak guide
fields the electric field caused by the spontaneous reconnection of the
oppositely directed magnetic field components points inward towards the X point and is nearly parallel to the reconnected magnetic field. However, in
the strong guide field it causes an Erec×Bg drift of the
magnetized electrons perpendicular to the guide and reconnection electric
fields which gives rise to the two asymmetric electron depletion regions
(exhausts or holes) and the electron accumulations at the boundaries of the
reconnection region. The accumulated electrons cannot leave to the
environment, however, being trapped at the boundary. They perform a partial
meandering motion in the crossed electric and magnetic fields and become
accelerated along the boundary of the reconnection site as indicated in
Fig. . This kind of heating and acceleration produces
momentum space anisotropies and deformations in the electron distribution
function which lead to the required perpendicular momentum space gradients.
Deformation of the initially isotropic electron distribution in the process
of reconnection has recently been shown in PIC simulations without the
presence of any guide field.
An example of this is reproduced in Fig. , adapted from
. It shows the rectangular exhaust region developing in the
absence of guide fields as a symmetric region extended along the reconnecting
magnetic field B0. The enormous dilution of plasma in this region is
striking. Nearly all the electrons are expelled in the course of the
reconnection from the X point, an effect we already saw in its asymmetric
form in the weak guide field simulation.
The lower two panels show on the left three mutually overlaid snapshots of
the momentum space electron distribution function at three different
locations in the exhaust during the evolution of the reconnection. It is
clear that the distribution function develops steep gradients mainly in
py. The ultimate distribution at the right end of the exhaust is seen in
the right panel. It exhibits an energetic ring distribution which strongly
resembles the model distributions used in the electron hole ECMI case
.
Simulation of particle density (a) in 2-D
non-guide field reconnection after. The about
rectangular shape of the electron exhaust is recognized as the green domain
with nearly no electrons in its centre. In this non-guide field simulation
the length of the exhaust extends to roughly 50λi on both sides of
X. More interesting is the evolution of the electron distribution
function (b, c). The left panel is a superposition of snapshots of
distribution functions at increasing distance to the left from the X point.
The distribution gradually evolves into the average distribution shown on the
right and exhibits a strong anisotropy in momentum space. The light green
ring with its yellow maximum shows the perpendicular anisotropy forming a
well-expressed ring structure (the quantitative colour scale is suppressed).
In the presence of a guide field the distribution forms a ring perpendicular
to the guide field. Without guide field such distributions excite the Weibel
instability
and create a secondary magnetic field which structures the current and leads
to magnetic turbulence. In the presence of a guide field Weibel instability is
suppressed while the anisotropy favours the ECMI.
Such a momentum distribution possesses moderate if not steep gradients in the
two perpendicular momenta perpendicular to the current (and thus to
the guide field). Moreover, like in the case of an electron hole, the ECMI resonance curve in momentum space is a large section of a circle along the yellowish
maximum in the distribution. It satisfies the necessary condition for
excitation of the ECMI under conditions of reconnection. The main difference
between these simulations and those with guide field will be found in a much
stronger asymmetry in the distribution function and therefore a deformation
of the resonance curve. However, we may in this model assume that for the
principal effect these asymmetries just represent technical complications in
performing calculations.
Exhaust properties of relevance for the ECMI 2
With these considerations and clarifications we have arrived at the point
where we can summarize the effects of reconnection which are of relevance to
the ECMI as follows:
Reconnection, whether or not proceeding in guide fields, generates
extended highly electron-diluted “exhaust” regions, which are elongated in
the direction of the reconnecting magnetic fields. With guide fields present,
these exhausts split into two asymmetrically located regions of diluted
electron populations. In addition, complementary regions of electron
accumulations evolve at the boundaries of the reconnection site.
The electron momentum distribution functions in and at the boundaries
of the exhaust become accelerated in a combination of the electric field that
is generated in the process of reconnection, and the cross-field drift of the
electrons in the strong guide field. This combined action causes a
deformation of the electron momentum space distribution in the plane
perpendicular to the guide field. It generates sufficiently steep
perpendicular electron momentum space gradients ∂fe(p⟂)/∂p⟂ on the electron distribution
function. These gradients exist in both perpendicular components of the
electron momentum, being the requirement for the ECMI. Acceleration of the
electrons in reconnection in general, whether in guide fields or not,
warrants the at least weakly relativistic nature of the electron
distribution.
The extension of the reconnection site and the electron exhaust along
the reconnection field and thus perpendicular to the guide field amount to
several ion-inertial lengths which corresponds to roughly 100 λe or
up to 103λD. Compared to electron holes, reconnection exhausts
are therefore macroscopic objects. Their extension along the current and
guide field cannot be read from any 2-D simulations but is at least of the
order of a few ion-inertial lengths. It will not exceed the parallel wavelength
λ‖≫λe of an IAW pulse propagating along the guide field.
Hence exhausts are 3-D objects of extensions of few to several λe in
each spatial direction and can have further substructure.
As a final observation one may note that the relation between the
generation of electron exhausts and IAWs implies that an electron exhaust,
once formed in reconnection, will necessarily propagate along the guide field
together with the IAW with the speed of a substantial fraction of the Alfvén
velocity VA based on the ambient plasma parameters. Because the IAW carries
the current, it propagates downward in the downward current region and upward
in the upward current region. One therefore expects similar propagation
directions for the exhausts.
Application to the ECMI
After these preparatory clarifications of our model we are now in the
position to apply the reconnection paradigm to the ECMI. In fact, the
similarity is striking between an electron exhaust in reconnection and an
electron hole produced by the nonlinear evolution of the Buneman instability
in unmagnetized as well as magnetized plasmas. Both are localized electron
depletions; both contain electron distributions with perpendicular momentum
space gradients and, under certain conditions, weakly relativistic electrons;
and both propagate along with the current flow. A schematic of the structure of
the exhaust in the normalized relativistic momentum space u=p/m
is given in Fig. .
The main differences between the model of an electron hole and the exhaust
are the different sizes. Electron holes are Debye-scale entities, while
reconnection exhausts are comparably very large electron inertial-scale
entities. This makes the latter indeed interesting candidates for hosting the
ECMI. Moreover, this similarity allows for a direct transfer of our ECMI
theory of radiation produced by electron holes to the ECMI radiation which
can be expected of being generated in a reconnection electron exhaust – with
the exhaust playing the role of an “electron quasi-hole”.
ECMI from electron holes/exhausts
Let us recall the properties of the ECMI in the case of electron holes
and write it in terms of the reconnection parameters.
For simplicity assume a circular hole.
The geometrical form of the
exhaust and thus the resonance curve could be modelled as ellipses which
would introduce eccentricity as another free parameter thus complicating the
calculation a bit. Though this is not too difficult to manage mathematically,
we ignore this complication because it does not provide any deeper insight
into the physics of the role reconnection plays in the ECMI.
Working in the
hole frame, the case also natural in exhausts, the resonance condition in
phase space is a circular section in perpendicular momentum space of radius
Rres=2(1-νc),νc=ω/ωc,
where now, because of the strong guide field Bg,
ωc=eBgm1+B02Bg2≈ωcg1+12B02Bg2.
The maximum growth rate is obtained at maximum resonance
Rm≈U-(Δu)2/U,U=VA/c,Δu=ve/c
and VA the drift velocity of the hole/exhaust, which in our case is the
Alfvén speed or a larger fraction of it based on the guide field, and ve
is the thermal speed of the hot trapped electrons, which is small with
respect to c but by no means small compared with the Alfvén speed. Then
Δu∼0.1, and the second term in Rm can be dropped. Moreover, we
have approximately
Mass-normalized momentum phase space of electrons in strong guide
field exhaust with u=p/m. The guide-field-aligned current
electrons have a broad electron distribution (similar to those observed in
the AKR downward current region) shown here along the u‖ axis which is
also the direction of the guide field Bg. The greenish region is the
highly electron-diluted phase space domain of exhaust electrons. This region
is filled with perpendicular drifting dilute electrons which move across the
hole and magnetic fields by E×Bg drift to the boundaries. The banana-shaped coloured region is filled with the exhaust electric field boundary
electrons which form non-symmetric beams at the boundary of the exhaust which
are directed parallel/antiparallel (depending on the part above or below the
exhaust X point) to the reconnected field-aligned current magnetic field
B0. It gives rise to a perpendicular phase space distribution
gradient ∂F(u)/∂u⟂>0 which is capable of
exciting the ECMI.
Rm≈U=mmiωcgωe.
The maximum normalized growth rate is obtained when using the full
expression () for Rm in the growth rate
see, which then becomes
ℑ(ω)ωcg|Rm≈απ34νcωe2ωc2(VAve)(cve)exp(-12ve2VA2),
where α≈Nh/Ne is the ratio of the exhaust density at the
depletion of electrons to the density of the environment, a number of order
α≲O (0.1) or so. This growth rate is inversely proportional
to the normalized frequency νc and thus decreases algebraically with
frequency (and the harmonics as well). In strong guide fields the
contribution of B0 is negligible, and the cyclotron frequency is solely
determined by the guide field Bg. The argument of the exponential is
ve2/2VA2=β(mi/m). In the IAW region where, according to our
discussion, reconnection becomes spontaneous and thus violent, β<m/mi,
and for the ambient plasma the exponential is between 1 and
∼1/e. However, the hot interior electrons have higher thermal
speed. They enter the growth rate, increase the interior β and make
VA≲ve, with the exponential well compensating for the factor
c/ve∼30. The normalized growth rate is therefore small. It must be
less than the cyclotron frequency. If we assume for the exponential
<3×10-4, we have the interior electron thermal to Alfvén speed
ratio ve/VA≳ 4, which yields
ℑ(ω)≲0.02ωcgνcωe2ωc2.
This is determined by the ambient ratio ωe2/ωc2 thus being a
substantial fraction of ωcg with fundamental emission close to and
slightly below ωcg. In the above-cited paper we estimated that the
frequency width below ωc is of the order of Δνc∼2βe2. Taking βe≲m/mi (as required for the IAW) implies
that the difference between the guide field cyclotron and the maximum
emission frequencies at the fundamental X mode is of the order of
Δνcg∼10-3, a small number. At ∼300 kHz in the AKR
this difference corresponds to ∼300 Hz. Emission is practically at the
guide field cyclotron frequency. Since the cyclotron period is short, there
is plenty of time to grow for the wave. The fundamental will be trapped
inside the exhaust for a long time, while higher harmonics can possibly escape,
depending on the ambient cyclotron-to-plasma frequency ratio.
Trapping of radiation inside the exhaust implies a number of other
interesting effects which have been discussed in relation to electron holes
. These include the possibility for
amplification due to sufficient time for experiencing many growth cycles,
quasilinear saturation of the radiation intensity, depletion of the
perpendicular momentum gradient in the electron distribution, interferences,
as well as other nonlinear effects at large radiation amplitude. One may
expect relatively high saturation levels of the trapped modes. Trapping also
implies convective radiation transport in the exhaust along the guide field
to locations where the plasma conditions become favourable for radiation
release.
Spatial displacement of ECMI source
The above growth rate also exhibits another observationally important effect.
It is restricted to the location of the exhaust in space. However, the
exhaust moves in two directions. Firstly it moves with the field-aligned
current along the guide magnetic field, an effect which is included in the
Alfvén speed VA and in the cyclotron frequency ωc. It also moves
at a slower speed perpendicular to the guide field shifting away from the
reconnection X point to either left or right along the reconnection magnetic
field (i.e. along the field-aligned current-sheet plane), an effect which,
when compared with VA, probably causes a weak modulation only. The main
modulation of the growth rate is caused by the spatial dependence of the
Alfvén speed along the guide field, when the X point moves with the
Alfvén wave along the field. The frequency dependence on location and thus
magnetic field is contained in ωc∼Bg, while νc∼1. Thus
the variation of the growth rate with coordinate s along the guide field is
determined essentially by the spatial variation of Bg(s)/N(s)
along the guide field as
ℑ(νc,s)∼N(s)/Bg(s).
In the case B(s)/N(s)≈ const, the growth rate behaves like
ℑ(νc,s)∼1/N(s). In addition the exponential factor affects
the growth rate if the Alfvén speed reduces, which happens only when the
guide field becomes very weak. Emission very close to the cyclotron frequency
or its harmonics will thus intensify with decreasing density. This
should be the situation in the upper auroral ionosphere for outward
propagation.
Discussion
The first condition for the ECMI is that there is more magnetic energy per
electron in the volume of the exhaust than the rest mass energy of an
electron:
ωe2ωce2=mec2B2/μ0N<1.
In this case it is clear that, as mentioned before, there are not enough
electrons to absorb any excess in free energy and carry it away in order to
establish thermodynamic equilibrium. In order to reduce the accumulated free
energy in the perpendicular velocity space gradient the plasma reacts by
directly exciting the free-space waves. In our case of reconnection the free
energy is the energy of the current, i.e. of the IAW. The process which
dissipates the energy is spontaneous collisionless reconnection. Radiation
eliminates just a fraction of the available reconnection energy. The main
dissipation proceeds via plasma heating and acceleration in reconnection.
Nevertheless, the conditions are in favour of direct excitation of radiation
via the ECMI. Thus, reconnection in strong guide fields will always be
accompanied by radio emission which is caused by the ECMI in the extended
electron exhaust. Its remote observation should provide information on
the plasma state (strength of magnetic field, density, plasma processes,
presence of reconnection, field-aligned currents, electron beams etc.).
At its fundamental the ECMI generates emission below but very close to
ωc based on Bg. Because the emission generated at location r1
cannot escape, the source where the emission is trapped must first move to a
location where the radiation frequency νc(r1)>νXc(r2) exceeds
the X-mode cut-off frequency, as illustrated in Fig. . For
an external observer any narrow frequency signal moving down or up in
frequency maps the local spatial variation of Bg that is related to the
spatial displacement of the electron exhaust along Bg to lower or
higher magnetic fields. The radiation will always be cut-off once the
emission frequency drops below the local X-mode cut-off (assuming it is
generated in the X mode almost perpendicular to the local field). Changing
emission frequency will thus have to be attributed to a motion of the
exhaust, which is restricted mainly to the direction along the field and thus
a consequence of the velocity of the IAW.
An observer inside the plasma will be unable to locate the radiation source
unless a measurement of the direction becomes possible. For such an observer
(spacecraft) any narrowband emission signals can both rise and fall in
frequency. Observed remotely, however, falling tones map the global
magnetic field variation with distance. Rising tones may be attributed to
Doppler-shift effects of an approaching source. Broad emission bands in
frequency imply that many emission sites exist simultaneously in an extended
source containing a spatially variable magnetic guide field such that many
places emit almost simultaneously. Such a band will always exhibit a low-frequency cut-off implied by the escape conditions.
Concerning emissions observed from the Sun, the ECMI model of reconnection
offers a wide field of possible applications. The solar atmosphere and corona
contain very strong magnetic guide fields, and are subject to field-aligned
current flow which are either carried simply by electron beams or also by
kinetic Alfvén waves. These are restricted to narrow flux tubes and can
reconnect across the ambient guide magnetic field. Such processes may lead to
all kinds of emissions in the ECMI and will always indicate reconnection
occurring.
Caveats and unresolved problems
Illustration of the requirements for escape of ECMI radiation
excited in the X mode to free space if the radiation is generated a frequency
beneath the electron cyclotron frequency of the guide field and moves from
R1→R2 with r0 distance of normalization. Excitation is on the left
at a frequency shown as the red dot. It occurs on the lower branch of the X mode. In a magnetic guide field of spatially decreasing strength (e.g. a
dipole field) from location r1<r2 to r2, the cut-off (index co), upper
hybrid (uh) and cyclotron frequencies (ce) drop below the emission frequency
νemc1 at r1. If the wave is able to tunnel the gap on the
way to the new location, its frequency νemci1 will be found on
the free-space branch of the X mode and can escape. Tunnelling is
non-trivial. It implies a rather severe condition on the radiated wavelength,
angle and density gradient. At r2>r1 emission is at the low indicated
frequency νemc2<νemc1 and is trapped in the plasma.
An outmoving source will emit downmoving tones. No upmoving tones are seen
remotely. They are inhibited by the condition of escape but can occur
inside the plasma.
The remaining problems concern the confinement of the radiation and its
escape to free space which is required for remote
observations.
These considerations are based on cold plasma theory
of the electromagnetic modes, the simplest case. It is clear that the
dispersion of all waves changes when the electron plasma becomes hot. Concerning the KAW, temperature increase implies larger gyroradii thus
favouring the transition to the inertial range. This means that the condition
β<m/mi is relaxed, and reconnection may set in already in the KAW
regime, in the auroral ionosphere at higher altitudes and weaker magnetic
fields. Higher temperatures also relax the escape conditions for the
fundamental which in reality might thus not be as severe as they appear in
our discussion. Higher harmonics, which are also excited by the ECMI though
at decreasing growth rate, should anyway have less problem escaping to the
environment and free space.
There is no problem for the O mode to escape if
excited. The fundamental X mode, on the other hand, cannot escape for the
obvious reason that it is excited by the ECMI at frequency
ω<ωce. Even in the case of the guide field this condition is a
severe restriction on the emission. If the guide field is strong, as is the
case close to a strongly magnetized planet, the X mode may escape from the
exhaust to the surrounding plasma but cannot leave from it to free space
unless it has moved out to a region where its frequency exceeds the X mode
cut-off ωXc, i.e. roughly the local upper hybrid frequency.
Still a problem remains as it has to bridge the no-propagation gap between
the upper hybrid and cut-off
ωuh≤ω≤ωXc frequencies. This
requires sufficiently steep density gradients roughly sharper than one
wavelength which is, however, a weak restriction only when the field is very
strong and the density low. With B0 the magnetic field at excitation of
the ECMI and Besc the field strength at escape, the X mode
escapes if the ratio of plasma to cyclotron frequency satisfies the condition
ωeωc,0<BescB012.
This bridging condition might not be too severe. It can be roughly reduced to
a condition on the emitted X-mode wavelength λX at escape:
λX>πdxdlnN(x)esc.
Here it is assumed that the attenuation of the radiation intensity when
crossing the stop-band drops much less than 1/e. A change of
density N by 1 order of magnitude then requires that it occurs over a
length shorter than Δx≲2λX/π.
More interesting than these well-known problems, which apply quite generally
to ECMI emission of radiation (but see footnote 5), is that the
exhaust is not fixed to a spatial location. Because it is a feature that
happens to take place in the IAW which propagates at the rather high Alfvén
velocity along the strong guide magnetic field, the exhaust propagates along
the magnetic guide field. Any slow deviation from the field direction caused
by the non-vanishing perpendicular wave speed, a fraction of VA, is
interesting though of little importance for the radiation. It may, however
contribute to direction-induced modulation and also Doppler shift in the
emitted frequency depending on the relative locations of radiation source
(exhaust) and observer (spacecraft, rocket). Effects of this kind have been
reported as typical for the AKR fine structure.
Conclusions
The ECMI has been discovered in auroral physics and was
proposed to explain the occurrence of the surprisingly intense terrestrial
kilometric radiation , later dubbed AKR and found to be
related to the occurrence of strong magnetic field-aligned currents, the
generic case of application of our theory.
Reconnection has so far not been brought into discussion in relation to the
ECMI. Instead, for a long time there was talk going on referring to “breaking
of field lines” in order to generate sufficient anomalous resistance along
the strong auroral geomagnetic field. In fact, as our model suggests, there
is no breaking of field lines. Field lines are an idealization of magnetic
flux tubes on scales exceeding the average electron gyroradius which carry a
magnetic flux
Φ=∫B⋅dF≈π〈rce〉2B,〈rce〉=mee〈ve⟂〉B,
where 〈ve⟂〉 is the ensemble average of the
perpendicular velocity of electrons, and 〈rce〉 is the
corresponding electron gyroradius. Reconnection is the process of rearranging
this flux on the scale below the electron gyroradius. This means
that on those scales identification of field lines at plasma temperatures
makes no sense (at temperatures far below plasma, field lines are defined as
flux tubes containing one elementary quantum flux Φ0=ℏ/e; these
never break but are exchanged in integer numbers).
Hence, instead of field-line breaking, the process that presumably takes
place is reconnection in the magnetic field of the field-aligned current
respectively the corresponding IAW that transports this current while the
guide field remains completely unaffected. In this process the energy stored
in the field-aligned current (carried by the IAW) is consumed, and by no means is
the guide field energy affected. The consumed energy is converted into
generation of the exhaust, the reconnection electric field, which points
perpendicular to the guide field, an Erec×Bg drift of
electrons, electron synchrotron acceleration when meandering, and ultimately
excitation of the ECMI in the exhaust. Unfortunately, no measurements of the
magnetic dissipation rate in the IAW have, at least to our knowledge, been
made or are available.
We thank the referees for bringing up this
important point. Though it would be highly desirable to directly measure the
dissipation rate in IAWs as it would provide information about the
possibility for reconnection in a real system like Earth's upper auroral
ionosphere, it is hard to imagine how such a measurement could be performed.
One way would be to think of tracing back the observation of IAW accelerated
electrons as, for instance, observed by to the action
of the reconnection-generated electric field. This, however, implies
reference to so far not performed PIC simulations of strong guide field
reconnection with KAW/IAW.
Recognition of this process as probably being fundamental to field-aligned
current systems in strong magnetic (guide) fields clarifies and rounds up the
picture of the generation of non-thermal radio radiation by the ECMI. It
identifies spontaneous magnetic reconnection as the process that takes place
not only in weak fields but also in strong guide fields whenever a
field-aligned current of width on the transverse electron inertial scale
flows along the guide field. This process of spontaneous reconnection in
strong current-aligned guide fields is particularly violent in collisionless
plasmas.
From this point of view one may realize that collisionless reconnection in
guide fields, whether weak or strong, is probably the best place for
application of the “electron-hole model” for the excitation of the ECMI.
The weak guide field model applies to almost all cases of reconnection going
on in nature, in particular also to magnetic turbulence which is believed to
dissipate via collisionless reconnection. The exceptions are ideally
symmetric non-guide field counterstreaming magnetized plasmas/magnetic
fields, and also the very strong guide fields carrying field-aligned currents
as investigated here. The latter will be restricted to strongly magnetized
stars, planets, and other strongly magnetized astrophysical objects, where
they should lead to very high radiation intensities. If this turns out to be
true, then the combination of collisionless reconnection and ECMI is a quite
general excitation mechanism of nonthermal radio radiation in the universe.
Summary and final remarks
The present paper suggests a mechanism as a working plan for an
efficient and realistic model of the ECMI under conditions of strong
current-aligned guide fields in collisionless reconnection or, otherwise,
collisionless reconnection in field-aligned current regions along strong
ambient magnetic fields. No attempt was made to develop a full quantitative
theory as this requires the performance of 2-D numerical PIC simulations
under conditions of very strong guide fields Bg≫B0, determination of
the electron exhaust region, fraction of plasma dilution, and – most
important – the microscopic weakly relativistic momentum space distribution
of the hot accelerated diluted trapped electron component in the exhaust. The
perpendicular momentum space gradients of this distribution are the free-energy source for the excitation of the ECMI at the fundamental and low
higher harmonics (and probably also the transverse electromagnetic electron
Bernstein modes). These results, which are obtained from the reconnection
simulations, should be used in the ECMI theory to numerically calculate the
growth rates for the cases of interest, based on realistic input parameters.
Performing this large program lies outside the presently proposed new model
of the ECMI in strong magnetic guide fields.
This mechanism combines two well established plasma processes: reconnection
and ECMI, is a local process working in comparably small regions which
explains the fine structure of radiation, and is of wide applicability. In
strong magnetic (guide) fields like planetary magnetospheres, the Sun, stars
and strongly magnetized astrophysical objects it may explain the generation
of drifting, highly time variable radio emission which maps the local
magnetic field strength along the drift path of the electron exhausts and
provides inferences about the plasma conditions and dynamics. In weak guide
fields the same mechanism should work causing low-frequency cyclotron radio
emission which contributes to energy loss. This is of substantial interest
also in large-scale magnetic/magnetohydrodynamic turbulence where the ECMI
radiation at the low electron cyclotron harmonics adds up from myriads of
small-scale reconnection sites to generate a stationary radio glow of the
entire turbulent region thereby bringing such large-scale turbulent regions
into radio-visibility, a so far not investigated effect of considerable
interest in astrophysical application.
In summary, ECMI in the electron exhaust/reconnection X-point region is a
promising and interesting phenomenon that may be realized quite frequently in
space and under cosmical conditions. When developing in strong guide field
reconnection related to field-aligned currents the emitted ECMI radiation may
become quite intense, resulting from localized drifting reconnection regions in
space.
No data sets were used in this article.
The authors declare that they have no conflict of
interest.
Acknowledgement
This work was part of a Visiting Scientist Programme at ISSI Bern. We thank
the ISSI Directorate for interest in this endeavour. We also thank
Raymond Pottelette for cooperation in the electron-hole radiation model, the
friendly ISSI staff for its hospitality, the ISSI computer administrator
Saliba Saliba for technical support, and the
librarians Andrea Fischer and Irmela Schweizer for access to the library and
literature. The very constructive comments of the referees are gratefully
acknowledged as they were of substantial help in clarifying the proposed
concepts. The topical editor, Elias
Roussos, thanks two anonymous referees for help in evaluating this paper.
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