1]Space Research Institute, Austrian Academy of Sciences, Graz, Austria
2]International Space Science Institute, Bern, Switzerland 3]Institute of
Geophysics, Munich University, Germany

_Correspondence to_<EMAIL_ADDRESS>
# ECMI Resonance in AKR Revisited: Hyperbolic Resonance, Harmonics, Wave-Wave
Interaction

W. Baumjohann  R. A. Treumann  [  [  [

###### Abstract

Recapitulation of the resonance condition for the fundamental and higher
electron cyclotron harmonics in the Electron Cyclotron Maser Instability
(ECMI) enables radiation below and confirms the possibility of radiation in a
narrow band above harmonics $n>1$. Near $n=1$ resonance on the confined lower
X-mode branch, amplification is supported by the decrease of phase and group
speeds. Confined slow large-amplitude quasi-electrostatic X-modes nonlinearly
modulate the plasma to form cavitons until self-trapped inside them at further
increasing wavenumber. They undergo wave-wave interaction, enabling escape to
free space in the second harmonic band below $n=2$. At sufficiently large
parallel wavenumber (oblique propagation), the fundamental resonance $n=1$ is
hyperbolic, a possibility so far missed but vital for an effective ECMI in the
upward current region. Here, the resonance hyperbola favourably fits the loss
cone boundary, the presumably important ECMI upward-current source-electron
distribution, to stimulate ECMI growth at available auroral electron energies.

###### keywords:

Electron Cyclotron Maser, Auroral Kilometric Radiation, Substorms, Harmonic
radiation, Resonance topology

##  1 Introduction

The Electron Cyclotron Maser (ECMI) instability mechanism (Twiss, 1958;
Hirshfield & Granatstein, 1977; Melrose, 1986; Treumann, 2006) has, for
already close to half a century (since Wu & Lee, 1979), been identified as the
canonical mechanism of generating AKR, the celebrated “auroral kilometric
radiation” (first analyzed by Gurnett, 1974, who identified its auroral
origin), a sporadic narrow-banded intense radio emission (discovered by
Benediktov et al., 1965, in hardly accessible work). It is emitted
preferentially during substorms, the major disturbance (Kamide & Baumjohann,
1993; Akasofu, 2004, 2021) of earth’s magnetosphere.

AKR (for a recent review of observations, cf., e.g., Yearby & Pickett, 2022)
propagates in the X-mode polarization and is radiated from the auroral zone
close to earth where the magnetic field is strong and the thermal plasma on
various scales becomes locally diluted, frequently with very small plasma-to-
cyclotron frequency ratio $\omega_{e}/\omega_{ce}\ll 1$. Under such
circumstances the ECMI sets on if only its _necessary_ and _sufficient_
conditions are both met simultaneously. These refer to the cyclotron resonance
of weakly relativistic electrons occupying an excited energy state (first
suggested in more general astrophysical context by Twiss, 1958) and, for the
sufficient condition, to a suitable electronic energy source. In classical
physics an excited non-thermal state is identified with some highly non-
thermal electron phase-space distribution function that deviates strongly from
a thermal (even anisotropic relativistic, see Treumann & Baumjohann, 2016)
Boltzmann-Maxwell distribution. Experimentally, the relevant distribution that
would be effective in exciting the ECMI has so far not yet been ultimately
identified. It is conventionally assumed to be kind of a simple (weakly
relativistic) loss-cone distribution (cf., e.g., Wu & Lee, 1979; Pritchett,
1984, and many others) in phase space as has frequently been measured (for a
typical measurement in situ the auroral magnetosphere, cf., e.g. Delory et
al., 1998) in the presumable AKR source region, an assumption which has
however been challenged (LaBelle & Treumann, 2002; Baumjohann & Treumann,
2022).

In a recent note (Baumjohann & Treumann, 2022) we inspected the paradigmatic
mechanism of auroral kilometric radiation, the most intense natural radio
emission in near-earth space. We noted a number of problems with that
mechanism which have not been or were at most approximately solved yet.
Nevertheless it seems to be certain that the ECMI is the sole sufficiently
strong, and thus reasonable and probable, mechanism that could explain the
emission of the surprisingly intense radiation pattern of AKR. Simultaneously
it may also serve as the non-thermal radio emission paradigm in strongly
magnetized objects for application in other places in space: planetary
magnetospheres in general, extrasolar planetary systems, and probably as well
in astrophysics to generate sporadic intense emissions in the radio band which
cannot be understood as synchrotron radiation but must have been caused by
unknown non-thermal electron distributions, even though the most effective of
these distributions still have to be identified unambiguously.

In the present note we do not go into the important question of the sufficient
condition: What kind of distributions are responsible for ECMI. We rather deal
merely with the necessary condition that has to be clarified first, prior to
worry about which most probable and experimentally supported phase-space
distribution must be chosen to obtain sufficient radiation intensities. This
is the question we attack here to be answered: Given _any suitable_
distribution function, under which conditions can the ECMI become excited and
escape into free space, hopefully at the observed large amplitudes? This
question is related to kinetic theory and the condition of resonance between
the ECMI-source electrons and the relevant electromagnetic wave mode.

##  2 Resonance – the necessary condition

The ECMI is a kinetic instability emitted in the X mode polarization (cf.,
eg., Melrose, 1986; Treumann, 2006). By reference to observations it
propagates (predominantly) in the electromagnetic X-R mode (Krall &
Trivelpiece, 1973; Baumjohann & Treumann, 1996) almost perpendicular to the
strong ambient magnetic field $\vec{B}$. In the ECMI the X mode is excited
when, as noted above, the electron component in the plasma exhibits a
particular momentum space distribution function, representing the equivalent
of an elevated energy state of the resonant electron component.

Assume a relativistic electron velocity (momentum) distribution
$f_{e}(\vec{p})$ of such an energetically elevated kind. In a certain range of
frequencies $\omega$ and wave numbers $\vec{k}$, it obeys the required
property and allows for the excitation of the ECMI with growth rate

$\Gamma(\omega,\vec{k})=-\mathrm{Im}\,\mathcal{D}(\omega,{\vec{k}})/\partial_{\omega}\mathrm{Re}\,\mathcal{D}(\omega,\vec{k})\big{|}_{\omega_{X}}=\sum_{n}\Gamma_{n}(\omega,\vec{k})>0.$
(1)

where $\omega(\vec{k})=\omega_{X}(\vec{k})$ is the X-mode frequency as
function of wave number $\vec{k}$, and $\mathcal{D}(\omega,\vec{k})$ is the
kinetic dispersion relation, explicitly given in (Krall & Trivelpiece, 1973;
Baumjohann & Treumann, 1996) and, for a loss-cone distribution, explicated in
(Wu & Qiu, 1983), taken at the X-mode frequency and wave number (for the full
magneto-ionic theory111It should be noted that the magneto-ionic theory is a
fluid theory. It assumes that those wave modes are present, while the hot
kinetic plasma component just serves its excitation or damping. In linear
dispersion this permits neglecting all nonlinear modifications. of the X mode
in a cold electron-proton plasma cf., e.g., Budden, 1988; Melrose, 1986).
$\mathcal{D}(\omega,\vec{k})$ results from phase space integration over the
source electron distribution function $f_{e}(\vec{p})$ in a homogeneous
magnetized plasma and its gyrotropic derivatives with respect to the
components $p_{\|},p_{\perp}$ of $\vec{p}$, a quite involved expression in a
non-Maxwellian plasma even for purely perpendicular wave propagation of the X
mode.

In instability each resonance $n$ provides a positive/negative contribution to
the total positive growth rate $\Gamma(\omega,\vec{k})>0$ at the particular
frequency. This growth rate, being a function of the X mode frequency
$\omega_{X}$, accordingly exhibits discrete maxima/minima at or near to each
resonance $n$, depending on the contribution of $\Gamma_{n}(\omega,\vec{k})$
being positive or not at that particular frequency. Plasmas like the upper-
auroral substorm-magnetosphere that are capable of exciting the ECMI with
growth rate $\Gamma>0$, are dilute with $\omega_{ce}\gg\omega_{e}$ and, for
the purposes of X-mode propagation can, sufficiently justified, be considered
cold and non-relativistic, as it is only the weakly-relativistic non-thermal
auroral electron population, being of different than ionospheric origin and
responsible for the unstable excitation of the ECMI. This may be different in
much hotter astrophysical plasmas, however, where the relativistic
modification in the X-R mode dispersion may have to be taken into account, a
problem we are not concerned about here in the context of the magnetosphere.
The assumed low density state relegates the source of AKR and excitation of
ECMI to the spatially extended auroral upward current region and generally
excludes the narrow and dense downward current region, which also implies that
the prevalent electron distribution is believed to be of the loss-cone family.
This notion has been challenged (Baumjohann & Treumann, 2022; Treumann et al.,
2012) though not disproved. In the following no attempt is made to calculate
the growth rate for any specific distribution, which leaves this question
open.

###  2.1 Harmonic resonances $n>1$

The necessary condition for contributing to instability at harmonic number
$n>1$ and parallel or antiparallel propagation

$\omega-n\omega_{ce}/\gamma-k_{\|}c\beta_{\|}=0$ (2)

with $k_{\|}=k\cos\theta$ is the $n$th resonance in the phase space integral,
with $n=\pm 1,\pm 2,\dots$, the cyclotron harmonic number which, in
calculating the total growth rate $\Gamma(\omega,\vec{k})$ as function of the
X mode frequency $\omega=\omega_{X}$, is summed over. Because the relativistic
factor $\gamma>1$, their fundamental (relativistic) resonant frequency
$\omega_{1X}=\omega_{ce}/\gamma$ at $n=1$ is always below the non-relativistic
cyclotron frequency $\omega_{ce}=eB/m_{e}$ and thus confined to the plasma
while in all higher harmonics $|n|>1$ the resonant frequencies $\omega_{nX}$
exceed the X mode stop band $\omega_{uh}\leq\omega_{X}\leq\omega_{co}^{u}$
between the upper hybrid
$\omega_{uh}=\omega_{ce}\sqrt{1+\omega_{e}^{2}/\omega_{ce}^{2}}$ and upper
cut-off
$\omega_{co}^{u}=\frac{1}{2}\omega_{ce}(\sqrt{1+4\omega_{e}^{2}/\omega_{ce}^{2}}+1)$
frequencies. While the fundamental cannot escape without help, the higher
harmonics would naturally radiate away at about the speed of light if excited,
having had little time for amplification and thus should generally be rather
weak.

In the growth rate sum, each term $\Gamma_{n}(\omega,\vec{k})$ gives the
contribution of the harmonic resonance $n$ to $\Gamma(\omega,\vec{k})$ or, in
other words, the growth rate of the harmonic frequency at number $n$. The
harmonic number $-\infty<n<\infty$ introduces an infinite chain of possible
resonances from which the given distribution function $f_{e}(\vec{p})$ and the
available resonant electron energies $\epsilon_{e}=m_{e}c^{2}\gamma(\vec{p})$
select. For emission, the AKR source must be relativistic (Wu & Lee, 1979;
Melrose, 1986, 1994; Treumann & Baumjohann, 1997), which is taken care of in
$\gamma(\vec{p})=\sqrt{1+p^{2}/m_{e}^{2}c^{2}}$, and the electron momentum
$\vec{p}=m_{e}\gamma c\vec{\beta}$, where $\vec{\beta}\equiv\vec{v}/c$ is
defined.

Working in momentum space $\vec{p}$ is more convenient. It has the advantage
that no restriction is to be made on $\vec{p}$ as the fully relativistic
momentum spreads the entire space $0\leq|p|<\infty$ thereby avoiding the
inconvenient upper bound on $\beta\leq 1$, the circle of light speed radius in
$\vec{\beta}$-space which apparently excludes any high-speed beams for
becoming involved. This artefact disappears in momentum space.

For any $\gamma\gtrsim 1$ and
$\kappa_{\|}=k_{\|}c/\omega_{ce}=kc\cos\theta/\omega_{ce}$ the fully
relativistic resonance condition reads

$\gamma(\vec{p})x-\kappa_{\|}p_{\|}/m_{e}c=n$ (3)

where $x=\omega/\omega_{ce}$ is the normalized frequency. Introducing the
parallel momentum shift $p_{0}=nm_{e}c\kappa_{\|}/(x^{2}-\kappa_{\|}^{2})$
yields the resonance line in gyrotropic momentum space $(p_{\|},p_{\perp})$
which is an ellipse

$\frac{(p_{\|}-p_{0})^{2}}{a_{\|p}^{2}}+\frac{p_{\perp}^{2}}{a_{\perp
p}^{2}}=1,\quad\mathrm{with}\quad 0<p^{2}<\infty$ (4)

where the two radii are given by

$\displaystyle a_{\|p}^{2}$ $\displaystyle=$ $\displaystyle
m^{2}_{e}c^{2}x^{2}(n^{2}+\kappa_{\|}^{2}-x^{2})/(x^{2}-\kappa^{2}_{\|})^{2}$
(5) $\displaystyle a^{2}_{\perp p}$ $\displaystyle=$ $\displaystyle
m^{2}_{e}c^{2}(n^{2}+\kappa^{2}_{\|}-x^{2})/(x^{2}-\kappa_{\|}^{2})$ (6)

For $\kappa_{\|}=k_{\|}=0$ the ellipse degenerates into an unshifted circle,
and all perpendicular harmonic emissions are below $x<n$ for any energy
$\gamma$. Usually the resonance ellipse is in the literature expressed in
$\vec{\beta}$. There is a subtle difference between the two representations of
$\vec{\beta}$ and $\vec{p}$ resonances. The former is insensitive to the
relation between frequency $x$ and parallel wavenumber $\kappa_{\|}$, whereas
in the latter this relation may affect the shape of the resonance curve, a
problem we will return to later.

Figure 1: The plasma-confined X-mode dispersion relation in the auroral upward
current AKR source region (lower solid curve) under the prevalent low-density
plasma condition $\omega_{e}\ll\omega_{ce}$. The upper curve is the free-space
X-mode. In the stop band between both X-mode branches the X mode cannot
propagate. Normalization of the frequencies is to $x\equiv\omega/\omega_{ce}$,
with $x=1$ the non-relativistic electron cyclotron frequency which is crossed
by the linear lower-branch dispersion curve slightly above the
relativistically allowed ECMI resonance. It propagates between the lower cut-
off and upper hybrid frequencies. The red line approaching $x=1$ is the
_nonlinear_ X-mode dispersion branch at maintained resonant frequency
$x_{\mathit{res}}\approx\mathrm{const}\lesssim 1$, stretching out to large
wave-numbers $k$. Note that here on the dotted line the linear dispersion (17)
is invalid. In this range both the phase and group velocities of the lower-
branch X mode at resonant frequency, which is held constant remaining below
$\omega_{ce}$, are substantially reduced, with the wave becoming quasi-
electrostatic in resonance with an appropriate distribution of electrons.
Wave-wave interaction becomes probable here and results in escaping radiation
at $x\lesssim 2$ on the free space upper X-mode branch. For 4-wave interaction
also higher harmonics may become excited. No deformation of the dispersion
relation as function of propagation angle $\theta$ is shown here, which would
slightly modify resonance, dispersion and interaction and under large
wavenumber conditions even switch from elliptic to hyperbolic (for the
hyperbolic case see below).

In oblique propagation the case $x^{2}>n^{2}+\kappa_{\|}^{2}$ is excluded, as
both radii would become imaginary. In order to maintain an ellipse, one
requires that $a_{\perp p}^{2}>0$ is positive, which implies that
$x>\kappa_{\|}$ and $n^{2}>x^{2}-\kappa_{\|}^{2}$. Of course, emission is
below

$x<\sqrt{n^{2}+\kappa_{\|}^{2}}$ (7)

The higher the harmonic number, the more oblique is the wave mode which
contributes. For small obliqueness $\theta\lesssim\pi/2$, the most interesting
case in the free space ECMI with $\omega_{Xn}=kc$, this sets a condition on
the angle of emission for extension above harmonic $n$ in the frequency band

$n<\frac{\omega_{Xn}}{\omega_{ce}}<\frac{n}{|\sin\theta|}$ (8)

which is a narrow range only. There is no restriction on any emission below
$n$, other than that the frequency must exceed the upper frequency cut-off
$\omega_{co}^{u}$. This gives for small ratio $\omega_{e}/\omega_{ce}<1$ the
auxiliary condition

$1+\frac{\omega_{e}^{2}}{\omega_{ce}^{2}}<\frac{\omega_{Xn}}{\omega_{ce}}=\frac{n}{\gamma}$
(9)

for excitation and propagation in the $n$th harmonic. An upper limit
$\gamma\lesssim n$ is set on the electron energy
$\epsilon_{e}=m_{e}c^{2}\gamma$ in resonance with the $n$th harmonic to
contribute to the radiation near the upper cut-off, meaning that all
resonances contribute to radiation in the lowest band of the free-space X mode
which, therefore, will be the most intense.

Below $n$ resonances and emissions are, in principle, unrestricted. However,
the last condition also allows just for a narrow range $|n|<x<\sqrt{2}\ |n|$
of frequencies above $|n|$, depending on the electron distribution function
and resonant energy $\gamma$ whether or not it favours ECMI. At higher
frequencies above this limit the resonance condition does not permit
radiation, a general necessary condition which is independent of any particle
distribution function. It weakly contradicts the claims in (Wu & Qiu, 1983)
while at the same time making them precise. Generally harmonic emissions will
always be below the harmonic $n$ with just that narrow range of oblique
emission above $n$. Observation of high-frequency bounded radiation can be
taken as measurement of $n$.

However, though harmonic resonances (and in case of causing instability
realized as radiation) are possible, they necessarily will be rather weak
because the radiation is on the upper X mode branch where it readily escapes
from the source region at light velocity $c$. The most intense directly
excited emission results from below the second harmonic
$x=\omega/\omega_{ce}<2$, just above the upper frequency cut-off

$\omega_{co}^{u}=\textstyle{\frac{1}{2}}\omega_{ce}\Big{[}\Big{(}1+4\omega_{e}^{2}/\omega_{ce}^{2}\Big{)}^{1/2}+1\Big{]}<\omega_{nX}$
(10)

where also all higher harmonics contribute if present. In the magnetosphere
only radiation below $n=2$ is of interest, being restricted to frequencies
$\omega_{co}^{u}<\omega_{2X}<2\omega_{ce}$ and yielding for the resonant
electron energy the upper limit

$1<\gamma<2/\big{(}1+\omega_{e}^{2}/\omega_{ce}^{2}\big{)}$ (11)

This implies that in the auroral AKR source region electrons of energy
$\epsilon_{e}=m_{e}\gamma c^{2}\lesssim 0.9$ MeV are eligible of directly
exciting AKR beneath the second X-mode harmonic in about perpendicular
propagation in free space, if only their distribution function suites the ECMI
instability. This range of necessary electron energies indeed includes auroral
energies suggesting that second harmonic AKR radiation can naturally originate
from the auroral magnetosphere, as was proposed by Wu & Qiu (1983) and
suggested from observation (Pottelette & Treumann, 2005). Radiation should
however be weak for the above mentioned reason of unrestricted escape and lack
of spatial amplification. Generally auroral electron energies are far below
this limit (however, for a different observation cf., Xiao et al., 2016). They
barely are capable of exciting higher harmonics. For this simple reason,
harmonics $n>2$ should be absent in AKR.

###  2.2 Ultra-relativistic resonance

The absence of any restriction on the momentum $\vec{p}$ suggests that even
very high resonant electron momenta/energies could in general participate in
the ECMI. Turning to the extreme, this raises interest in the ultra-
relativistic limit, which here is listed for completeness, not for application
in the magnetosphere but probably be of vital importance in astrophysics.

Ultra-relativistic conditions imply $p^{2}\gg m_{e}^{2}c^{2}$ and
$\gamma(\vec{p})=p/m_{e}c$. The resonance ellipse has radii

$\frac{a^{2}_{\|p}}{m_{e}^{2}c^{2}}=\frac{n^{2}x^{2}}{(x^{2}-\kappa_{\|}^{2})^{2}},\quad\frac{a^{2}_{\perp
p}}{m_{e}^{2}c^{2}}=\frac{n^{2}}{x^{2}-\kappa_{\|}^{2}}$ (12)

It degenerates into a circle in perpendicular propagation $\kappa_{\|}=0$ of
radius $a_{p}=nm_{e}c/x$. Generally one has

$\frac{\omega}{\omega_{ce}}=\frac{n}{p/m_{e}c}+\kappa_{\|}\cos\theta_{p}$ (13)

where $\theta_{p}$ is the angle of the momenta of the resonant particles. The
bounds on either $n$ or $p$ are obtained when replacing the frequency with the
upper X-mode cut-off

$\frac{|n|}{p/m_{e}c}>\big{|}1+\frac{\omega^{2}_{e}}{\omega^{2}_{ce}}-\kappa_{\|}\cos\theta_{p}\big{|},\quad|n|>1$
(14)

Remember that $n$ can be both positive and negative, as it enters the sum of
harmonic growth rates $\Gamma_{n}$. The second condition on $n$ just points
out that this limitation applies to all harmonic emissions propagating in the
free space X-R mode.

###  2.3 Resonance at the fundamental

In view of application to the terrestrial magnetosphere it has been correctly
claimed (Wu & Lee, 1979) that the (weakly relativistic) ECMI resonance
dominates at harmonic number $n=1$, the fundamental. The reasons for this we
have already listed: Any higher harmonic has barely time for being amplified
substantially before leaving the source. Therefore the apparent theoretical
(Wu & Qiu, 1983) and suspected or real observationally claimed presence of any
intense second harmonic AKR emission in the magnetosphere can hardly be
understood as direct excitation.

Restricting to resonance on the lower X-mode branch eliminates all harmonics
$n>1$. With $0\leq\gamma-1=\sqrt{1+p^{2}/m^{2}c^{2}}-1\ll 1$ this is true in
the auroral region, from where the ECMI wave cannot escape into free space
(Fig. 1), as this requires traversing the X-mode stop band
$\omega_{uh}\leq\omega\leq\omega_{co}^{u}$ between the upper hybrid and upper
cut-off frequencies.

Resonance is generally not restricted to proximity to $\omega_{ce}$. The lower
X-mode branch for zero wave number $k\approx 0$ starts at the lower cut-off
frequency
$\omega_{X}>\omega_{co}^{l}=\frac{1}{2}[(\omega_{ce}^{2}+4\omega_{e}^{2})^{1/2}-\omega_{ce}]\approx\omega_{e}^{2}/\omega_{ce}$
for small ratios of plasma-to-cyclotron frequency, the parameter range where
the ECMI becomes effective. Assuming strictly perpendicular propagation
$k_{\|}=0$ all relativistic factors

$1<\gamma<\omega_{ce}^{2}/\omega_{e}^{2}$ (15)

of electrons in the distribution function could in principle participate in
the resonance. The oblique relativistic resonance condition for those
electrons is given by (3) with $n=1$. The parallel shift (now conventionally
expressed in $\vec{\beta}$) becomes
$\beta_{0}=\kappa_{\|}/(1+\kappa_{\|}^{2})<1$, and the elliptic radii are

$\frac{a_{\|}^{2}}{m_{e}^{2}c^{2}}=\frac{1+\kappa^{2}_{\|}-x^{2}}{(1+\kappa_{\|}^{2})^{2}},\quad\frac{a_{\perp}^{2}}{m_{e}^{2}c^{2}}=1-\frac{x^{2}}{1+\kappa_{\|}^{2}}$
(16)

With large $\gamma\lesssim\omega^{2}_{ce}/\omega^{2}_{e}$ the resonance is
placed far below $\omega_{ce}$ on the lower X-mode branch by the factor
$\omega_{e}/\omega_{ce}$ in frequency, if only sufficiently high energy
electrons would be available, implying that the resonant energy in this case
would be high. In the magnetosphere we have
$\omega_{ce}^{2}/\omega_{e}^{2}\gtrsim 10^{2}$ or even larger which, well away
from the cyclotron frequency, yields $\epsilon_{e}=m_{e}\gamma c^{2}\sim 50$
MeV. Such (about ultra-relativistic) electrons are absent there which, in
retrospect, justifies the assumption of proximity of the resonance to
$\omega_{ce}$. (We note, however, that high energies have been referred to by
Xiao et al., 2016, for apparently observed (A)KR whose source the authors
attribute to the radiation belts.) In remote astrophysical systems, on the
other hand, containing very-high, mostly even ultra-relativistic electrons,
proximity to the cyclotron frequency may become spurious, and excitation of
lower branch trapped long wavelength $k_{\perp}\gtrsim 0$ X-mode waves near
the lower cut-off $\omega^{l}_{co}<\omega_{X}^{l}\ll\omega_{ce}$ may become
possible or should be the rule. There the ECMI would include almost the entire
lower X-mode branch.

A broad energetic electron distribution may undergo resonance over a large
part of the lower X-mode branch contributing to the ECMI and growth of lower
branch X-mode waves in a wide range of frequencies
$\omega_{X}^{l}<\omega_{X}(k)<\omega_{ce}$. The wave numbers in resonance are
obtained from the pure lower branch electron X-mode dispersion relation in
perpendicular propagation (Krall & Trivelpiece, 1973; Baumjohann & Treumann,
1996)

$\frac{k^{2}c^{2}}{\omega_{ce}^{2}}=\frac{(\omega_{co}^{u2}-\omega^{2})(\omega^{2}-\omega_{co}^{l2})}{\omega_{ce}^{2}(\omega_{uh}^{2}-\omega^{2})}$
(17)

where for our purely electronic purposes the ion contributions as well as
higher powers of small quantities have been neglected.

This dispersion curve crosses the cyclotron frequency at $\omega=\omega_{ce}$,
corresponding to $k_{m\perp}\approx\sqrt{2}\ \omega_{e}/c$ which is about the
inverse electron inertial length $\lambda_{e}=c/\omega_{e}$, setting an upper
limit on the resonant wave number $k_{\perp}$, a large number respectively
short wavelength indeed. The unstable range $\Delta k_{\perp}$ maps to the
energy range $\Delta\gamma(k_{\perp})$ of resonant electrons

$0<k_{\perp}(\omega,\gamma)\ \lesssim\sqrt{2}/\lambda_{e}$ (18)

Depending on the electron distribution function, instability becomes possible
almost everywhere along the lower X mode branch if only sufficiently large
electron energies are available. $k_{\perp}=0$ corresponds to reflection at
the lower X-mode cut-off $\omega_{co}^{l}$ and is clearly outside the unstable
wavenumber range.

Near $\omega\approx\omega_{ce}$ the phase velocity of the X mode becomes
$v_{X}\approx c/\sqrt{2}$ implying that at an observed frequency
$\omega\lesssim\omega_{ce}\approx 300$ kHz, the nominal magnetospheric AKR
frequency (Gurnett, 1974), the phase speed is low enough for the wave to
experience several amplifications before leaving the source.

In order to check this, we obtain for the group velocity

$v_{gX}=\frac{d\omega}{dk_{\perp}}\approx-2^{3/2}\Big{(}\frac{\omega_{e}}{\omega_{ce}}\Big{)}^{2}c\ll-c$
(19)

which is much smaller than the velocity of light. For an ECMI growth rate
$\Gamma(\omega_{X})$ the spatial amplification rate $K\approx-\Gamma/v_{gX}$
becomes

$K(\omega)\approx\frac{\Gamma(\omega)}{2\sqrt{2}c}\Big{(}\frac{\omega_{ce}}{\omega_{e}}\Big{)}^{2}$
(20)

a comparably large value. Assume $\omega_{e}/\omega_{ce}\lesssim 0.1$, as
approximately applicable to the auroral source region, the group velocity is
just the order of $v_{gX}\lesssim 10^{3}$ km/s, in the auroral magnetosphere
this compares to the Alfvén velocity $v_{A}$, giving roughly $K\gtrsim 2\times
10^{-4}\Gamma(\omega_{X})$ km-1. So, for a growth rate $\Gamma\sim
10^{-4}\omega$, we have at the nominal emission frequency of $\omega\sim 300$
kHz that

$K\gtrsim 4\times 10^{-1}\leavevmode\nobreak\ \mathrm{km}^{-1}$ (21)

This corresponds to several e-folding lengths over one wavelength, sufficient
for excitation of moderate (or even large) amplitude X-mode waves. Of course,
growth rates $\Gamma(\omega)$ of this order may be extreme. Reducing
$\Gamma(\omega)$ by two orders of magnitude still gives one e-folding over one
wavelength of $\lambda_{X}\sim 1$ km within the oscillation and growth times.
Such amplification rates are still large.

This estimate suggests that the ECMI on the lower X-mode branch is capable of
generating rather large amplitude confined X-mode waves close to the local
non-relativistic cyclotron frequency $\omega_{ce}$ predominantly because there
the group velocity is strongly reduced, and the wave has sufficient time to
grow before leaving the source.

Growth becomes substantial here if the source is sufficiently extended over a
number of wavelengths in the direction of propagation about perpendicular to
the magnetic field $\vec{B}_{0}$. This very fact makes the ECMI on the lower
X-mode branch (see Figure 1) very interesting not just by itself but also for
its possible nonlinear interaction with the dilute plasma component whose
pressure is necessarily low and thus susceptible to large amplitude
electromagnetic radiation and the implications of its pondero-motive force.
Modulation instability of the unstable ECMI may become possible causing chains
of solitons and structure in the plasma, effects which have not yet been
considered in connection with ECMI but may play an important role if
investigating its fine structure (compare Figure 2 in Baumjohann & Treumann,
2022) and the reaction/response of the auroral environment to its presence.

Figure 2: The different resonance topographies Eq. (5) in dependence on the
normalized frequency $x=\omega/\omega_{ce}$ in the fundamental band $x<1$.
Elliptic resonance curves of the sort shown in the left part of the figure
occur to the left of the line $x=\kappa_{\|}$ and below the curve
$x^{2}=1+\kappa_{\|}^{2}$ in the range $x<1$ required by the fully
relativistic resonance in the fundamental band $n=1$. Hyperbolic resonance is
obtained in the yellow part on the right for $x<\kappa_{\|}$ and $x<1$.
Resonant hyperboles are the kind indicated on the left in red colour.
Apparently their range is substantially larger than that of the ellipses. But
its extension depends on the restrictions on the large wave numbers that
emerge from in the nonlinear evolution of the resonance. The range of
resonance is in fact not restricted to proximity to $x=1$ as in the weakly
relativistic resonance theory. At higher relativistic energy with $\gamma\gg
1$, frequencies well below $x=1$ become resonant.

Away from the local cyclotron frequency $\omega_{ce}$, however, growth is
reduced on the X-mode branch where higher energy electrons $\gamma\gg 0.1$ are
required to be present in an excited state, a case barely existing in the
auroral region. Here the group speed increases slightly before dropping to low
values when approaching the lower X-mode cut-off. In the auroral magnetosphere
large numbers of relativistic electrons are absent (cf., however, Xiao et al.,
2016, for a counter example). The relation between $k_{\perp}$ and $\gamma$ is

$k_{\perp}^{2}\lambda_{e}^{2}+1\approx\frac{1}{\gamma^{2}}\frac{\omega_{ce}^{2}}{\omega_{e}^{2}}$
(22)

In the longer-wavelength regime $\lambda_{\perp}>\lambda_{e}$ one has from
here

$1<\gamma\lesssim\frac{\omega_{ce}}{\omega_{e}}\big{(}1-{\textstyle\frac{1}{2}}k_{\perp}^{2}\lambda_{e}^{2}\big{)}$
(23)

between $k_{\perp}$ and $\gamma$ away from the plasma resonance with resonant
frequency reduced due to the increase in the required resonant energy. These
wave numbers define the location of the resonance on the lower X-mode branch
according to (17).

The elliptical nature of the resonance condition is however not easily matched
by any of the reasonable and measured electron distribution functions in the
magnetosphere. Such a distribution should be some kind of a (phase space
shifted) _hollow_ beam or _horseshoe_ distribution with parallel shift
$\beta_{0}$ (in terms of the momentum $p_{0}$), highly diluted regions lacking
higher-energy electrons in the hole. Their boundaries are formed by
accelerated/scattered cold electron beams (Muschietti et al., 1999; Ergun et
al., 1998). This should provide the required positive $\beta_{\perp}$-space
gradient mimicking an excited state of the resonant electrons. The beam-nature
of the distribution is imposed by the finite displacement $\pm p_{0}$ along
the external field which is seen in the observer’s frame. This poses the
question of the formation of such hollow beam distributions. Reasons for them
can be found in the generation of electron holes by strong field-aligned
currents and have been attributed to the downward current region in the
auroral magnetosphere (Treumann et al., 2012) where such partial hollows have
been assumed, either being electron holes or if of larger scale so-called
horse-shoes.

Guided by auroral zone observations it however seemed, and from theory was
also supported that the prevalent electron distribution of down going
electrons, part of which is reflected, part absorbed by the dense ionosphere
below, would become a loss-cone distribution. Numerical calculations based on
the resonance ellipse and using observed loss cones (Omidi & Gurnett, 1984)
found just very small amplification rates there. Moreover, from VLF
observation and theory it became pretty clear that any loss cones are readily
depleted or strongly reduced by intense interaction and amplification of VLF
noise (LaBelle & Treumann, 2002). The latter leaves little rudimentary
resonance for the excitation of the ECMI and agrees well with the calculations
based on the loss-cone.

Other relevant electron source-distributions are caused in the generation of
electron holes. These result in strong field aligned currents which decay into
localized electrostatic structures of few Debye length
$\ell_{\|}\sim\mathrm{few}\ \lambda_{D}$ extension along the magnetic field
but large perpendicular scale $\ell_{\perp}\gg\ell_{\|}$ of the order of the
perpendicular X mode wavelength $\ell_{\perp}\sim k_{\perp}^{-1}$. Inside
those holes the conditions of excited electron states are satisfied. Here we
do not go into a detailed investigation of this most interesting fact
(Muschietti et al., 1999) to that we had referred in earlier work (Treumann et
al., 2012). In the next section we deal with the question which interests us
here most: the condition of excitation of the ECMI on the lower X-mode branch
and its possible escape into free space without requiring any usually evoked
propagation effects in inhomogeneous plasmas (Zarka et al., 1986; Louarn et
al., 1990; Lamy et al., 2010).

Depending on the available electron energy and the excited phase space
distribution function $f_{e}(p_{\perp},p_{\|})$, the ECMI will become
effective over a large section of the lower-X mode branch to drive the X mode
unstable. The range where this could happen is prescribed by the condition of
a dilute plasma with $\omega_{e}^{2}\ll\omega_{ce}^{2}$ and the additional
necessary condition on the distribution function that $\partial f_{e}/\partial
p_{\perp}>0$ in an appropriate range on the resonance curve in the
$(p_{\|},p_{\perp})$-plane. This resonance line for purely perpendicular
propagation is given by

$\sqrt{1+p_{\perp}^{2}/m^{2}c^{2}+p_{\|}^{2}/m^{2}c^{2}}=\omega_{ce}/\omega^{l}_{X}(k_{\perp})$
(24)

with $\omega^{l}_{X}(k_{\perp})$ the lower-X mode branch frequency with the
right-hand side a function of $k_{\perp}$. Any of those unstable X-mode waves
have of course frequency $\omega^{l}_{X}<\omega_{ce}$ below the
nonrelativistic cyclotron frequency. They are confined to the plasma and
detectable only _in situ_ , unless a mechanism is found which allows them to
either escape into free space by tunnelling the ECMI-X-mode stop-band
$\omega_{ce}<\omega<\omega_{co}^{u}$ between the cyclotron and upper X-mode
cut-off frequencies.

The other possibility is to undergo a three-wave interaction process as we
proposed (Baumjohann & Treumann, 2022) and will be investigated below.
However, staying with strictly perpendicular propagation one may also envisage
direct excitation of higher harmonics $n>1$ of the cyclotron frequency
$\omega_{X}^{u}=n\omega_{ce}/\gamma$. Before returning to the three wave
process, we briefly discuss such

###  2.4 Wave-wave interaction at the fundamental

From observations is not entirely clear whether AKR is observed in the second
harmonic or not. Fundamental radiation is trapped below the X-mode stop band
(see Figure 1 ) and escapes only under strongly inhomogeneous conditions which
have not yet been clarified convincingly. On the other hand, harmonic
radiation when identified appears to be surprisingly intense. For the above
noted reasons one may doubt in the importance of any direct radiation
mechanism. The large growth rates $\Gamma_{n}\sim n\omega_{ce}$ respectively
large amplification factors $\kappa_{n}$ they require are hard to reproduce in
theory. Free space modes escape quickly before picking up any amplification
unless the source extends over very large distances while maintaining all the
conditions in favour of excitation. In addition to the obvious confinement of
the fundamental $\omega_{X}\lesssim\omega_{ce}$ observation of apparently
intense harmonic radiation (cf., e.g., Pottelette et al., 2001) in the
spectrograms poses a problem.

Of course, these conditions refer, in addition, to the required sufficient
condition on the electron distribution function in phase space and are,
probably, not very easy to satisfy in general. They require rather special
electron distribution functions which, in order to contribute efficiently to
growth, must adapt to the particular geometry of the resonance curve in phase
space. Such distribution functions must possess a positive perpendicular
momentum gradient along the phase space resonance or at least large parts of
it in order to pick up the contributions of as many electrons along the
resonance as are elevated to the higher momentum/energy excitation level.

Here we are interested in conditions which allow transformation of the ECMI at
the lower X-mode branch into the free space mode. As argued above, at the
lower X-mode branch the ECMI can indeed attain large amplification because of
the comparably slow group velocity under the conditions below
$\omega^{l}<\omega_{ce}$. Since waves can be excited here in any direction
perpendicular to the external magnetic field, a three wave process (Sagdeev &
Galeev, 1969; Davidson, 1972) suggested in (Baumjohann & Treumann, 2022),
becomes possible where two lower branch modes interact to compensate their
large perpendicular wave numbers and result in a long wavelength mode at
roughly the sum of the two frequencies according to the three-wave interaction
conditions of conservation of total energy and momentum

$\omega^{u}_{X}=\omega^{l}_{1X}+\omega^{l}_{2X},\quad
k^{u}_{\perp}=k^{l}_{1\perp}-k^{l}_{2\perp}\ll k^{l}_{1,2\perp}$ (25)

It generates a long wavelength $k_{\perp}^{u}\approx\omega_{X}^{u}/c$ upper
branch X mode of frequency $\omega^{u}_{X}\lesssim 2\omega_{ce}$ below though
near the second harmonic $n=2$. This wave is large amplitude, propagates in
the free space mode, can escape without any difficulty from the source region
and does not need any further amplification. The condition under that it may
meet the free space branch is that its frequency must be above the upper
X-mode-branch cut-off. Since each of the resonant waves on the lower branch
satisfies the resonance condition $\omega_{X}^{l}=\omega_{ce}/\gamma$ the
condition for escape of the ECMI-wave at the upper branch becomes

$\frac{\gamma_{1}+\gamma_{2}}{\gamma_{1}\gamma_{2}}>1+\omega_{e}^{2}/\omega_{ce}^{2}$
(26)

where the indices refer to the two lower branch waves that participate in the
interaction, and we neglect powers of small quantities. If the resonant
energies do not differ much, this yields

$1<\gamma<\frac{2}{1+\omega_{e}^{2}/\omega_{ce}^{2}}$ (27)

which of course is the same condition as for direct excitation below the
second harmonic $n=2$ on the upper X mode branch given above. Its meaning in
the present case is, however, quite different. The two interacting waves are
slow and have grown to large amplitudes. In head-on collisional interaction
they result in large-amplitude escaping radiation just below the second
harmonic $n<2$. For small frequency ratio $\omega_{e}/\omega_{ce}$ in the
denominator, the marginal relativistic factor is close to $\gamma\lesssim 2$
or, as already noted, the upper limit on the electron energy is
$\epsilon_{e}\lesssim 0.9$ MeV.

Low energies excite the ECMI on the lower X-mode branch close to
$\omega_{ce}$, but higher energies near the upper energy limit do also
contribute. They shift the resonance down on the lower branch according to
$\omega^{l}_{X}=\omega_{ce}/\gamma$. Thus the ECMI can, in principle, become
excited almost along the entire lower X-mode branch. Escaping radiation
generated via the three-wave collision process is nevertheless allowed only
for those waves excited by electrons of the above limited energies. Their
energy determines the location of the resonance on the lower X-mode branch
dispersion relation (17).

We have $\omega^{2}\ll\omega_{co}^{u2}$ and neglect higher powers of
$\omega_{e}/\omega_{ce}$, which is justified in all cases where the ECMI is
expected to become effective. In that case the lower X-mode branch dispersion
relation (17) simplifies yielding

$\frac{k_{\perp}^{2}c^{2}}{\omega_{ce}^{2}}\approx\frac{\omega^{2}-\omega_{co}^{l2}}{\omega_{ce}^{2}}=\frac{1}{\gamma^{2}}-\frac{\omega_{e}^{2}}{\omega_{ce}^{2}}$
(28)

for the relation between $k_{\perp}$ and $\gamma$, which reproduces condition
(15) while holding all along the lower X-mode branch

$1<\gamma<\frac{\omega_{ce}/\omega_{e}}{\sqrt{k_{\perp}^{2}\lambda_{e}^{2}+1}}$
(29)

All wave numbers

$k_{\perp}\lambda_{e}\lesssim\frac{\omega_{ce}}{\omega_{e}}\Big{(}1-{\textstyle\frac{1}{2}}\frac{\omega_{e}}{\omega_{ce}}\Big{)}$
(30)

are eligible for the three wave ECMI emission process above the upper cut-off
frequency, including almost the entire lower branch wave numbers. Combination
of the three last conditions on $\gamma$ and $k_{\perp}$ yields that the
three-wave interaction will become effective for frequency ratios

$\omega_{ce}>\sqrt{3}\ \omega_{e}$ (31)

a condition that is easily met in the auroral magnetosphere where under the
conditions when emission of AKR occurs, the electron cyclotron frequency by
far exceeds the plasma frequency.

To close this section, it is of substantial interest noting that exactly the
same nonlinear wave-wave interaction mechanism of generation of radiation in
the free-space X mode has been explicated in detail for the Z mode (Yoon et
al., 2016) in the approximate second $n=2$ and fourth $n=4$ harmonics of the
cyclotron frequency. This mechanism would in principle compete with the ECMI
wave-wave generation mechanism on the lower X mode branch proposed here, if
not the two mechanisms would unfortunately exclude each other. The ECMI works
solely under the condition that the ratio $\omega_{e}/\omega_{ce}\ll 1$ is
small. In this case the Z mode becomes confined to a rather narrow region
around $x\approx\omega_{e}/\omega_{ce}$ (Melrose, 1986). On the other hand, in
the opposite situation when this ratio is large, as presumably for instance in
the downward current region, then the ECMI becomes unimportant and excitation
of the Z mode dominates (though see the discussion below in the section on
downward currents). This is a very interesting case, indeed, as it predicts
that in that case AKR could be radiated from nonlinear coalescence of Z modes
near the upper hybrid frequency $\omega\sim\omega_{uh}$. Generation of Z modes
in that case is much less restricted than the ECMI on the X mode. It is well
known that the Z mode is generated rather frequently by the continuous
presence of VLF (LaBelle & Treumann, 2002) in the auroral magnetosphere where
it is not necessarily found just in the auroral zone and at extreme magnetic
activity but spreads over a wider latitudinal and longitudinal region in space
and in time. One expects that by the mechanism of wave-wave interaction in
both, Z modes (possibly in the downward current region outside the upward
current region) and X modes (preferentially in the upward current region),
produce intense escaping AKR at nearly same harmonic frequencies in the free-
space X mode.

Figure 3: Measured loss-cone distribution in the presumable AKR upward current
auroral source region (after Delory et al., 1998, which should be consulted
for a detailed description of the observations). The intensity of downward
electron fluxes is colour-coded (from black to red covering 5 orders of flux
magnitude) with red highest intensities, here indicating the denser warm
auroral-magnetospheric electron background. The comparably wide upward
(negative normalized parallel velocities $\beta_{\|}=v_{\|}/c<0$) electron
loss-cone is well exhibited. Circles show its theoretical (solid) and observed
(dashed) low-speed bounds. At positive parallel speeds the red banana
indicates the high energy downward auroral electron beam. The two black
hyperbolas are theoretical angular limits of inner and outer phase space
regions. The dotted red hyperbola at negative (upward) $\beta_{\|}$ is the
resonant hyperbola about along the loss-cone boundary with $k_{\|}c>\omega$.
Positive perpendicular gradients along and inside this hyperbola contribute to
resonant ECMI growth. The loss cone is wide enough to host a continuity of
such hyperboles inside its bounds thus providing the ECMI a bandwidth
$\Delta\omega$ at constant frequency $\omega$. Uncertainty of the observations
is too large for calculating quantitative numbers.

###  2.5 Nonlinear evolution

The ECMI-unstable lower X-mode branch beneath $\omega_{X}/\omega_{ce}\equiv
x=1$ is a slow wave with comparably low phase velocity $v_{ph}=\omega/k<c$
and, in particular, low group speed $v_{g}=\partial\omega/\partial k<c$, both
substantially less than the velocity of light. As argued in the previous
section, these properties allow the wave to grow and achieve amplitudes which
are large enough to violate the linear assumption. In this moderately or large
amplitude state and being unable to leave the plasma below the X-mode stop
band, the wave exerts a slowly variable ponderomotive force on the dilute low-
pressure plasma background. (Note that the plasma is generally assumed to be
at most temperate in this entire theory; what concerns the wave dispersion, so
the background is even assumed to be cold.) The ponderomotive force results
from an average over the entire ensemble of unstable waves thus being the
effect of collective action. The variation of the ponderomotive force is slow;
it occurs on the ion time scale. The plasma reacts to this force as described
by the Sagdeev-Zakharov (slowly variable nonlinear Schrödinger) equation (cf.,
e.g. Davidson, 1972; Treumann & Baumjohann, 1997) by exciting ion acoustic
waves which under stationary conditions lead to envelope solitons, density
depletions that trap the high frequency lower branch X-mode waves. The
relative quasi-neutral density modulation $\delta N/N_{0}$ (amplitude of the
ion-acoustic wave) of these structures, which from pressure balance are
density depletions (cavitons), is related to the X-mode amplitude as

$\frac{\delta
N}{N_{0}}\approx-\frac{\epsilon_{0}}{4m_{i}N_{0}c_{ia}^{2}}|\delta E_{X}|^{2}$
(32)

with $c_{ia}\ll c$ the ion acoustic speed. Trapping of the X-mode inside the
density depletions causes a spatial modulation of the wave spectrum. (One may
note that only the electric field amplitude appears here. The magnetic
amplitude of the electromagnetic wave is relativistically small.) At the same
time it leads to two other important effects.

Firstly, trapping of the spectrum of participating X modes splits the waves
into two populations of oppositely directed wave numbers $\pm k_{tr}$ which
bounce back and forth inside the density depletions. Several of these waves
will thus participate in the above described three-wave process to cause
second harmonic radiation $\omega_{X}\lesssim 2\omega_{ce}$. This leads to
losses of wave energy and may ultimately terminate the deepening of the
density depletions, i.e. stabilising the nonlinear ion-acoustic wave
amplitude, even though radiative losses generally remain only weak.

The second interesting effect concerns the trapped wave mode wavelength
$\lambda$ (or wavenumber $k_{X}$). Conservation of caviton shape over the time
of caviton formation and evolution implies that the number of waves
(wavelengths) inside the cavity is conserved and remains constant. Shrinking
the cavity size thus necessarily shortens the wavelengths and increases the
wavenumber $k_{X}$ according to

$k^{2}\propto\epsilon_{0}|\delta E_{X}|^{2}$ (33)

with the right hand side the wave energy density which evolves during
nonlinear formation of the caviton, i.e. the sum of the individual wave
energies $\hbar\omega$ per volume of the caviton with frequency being
conserved. This affects the dispersion of the trapped waves. This is very
interesting by itself.

Though the trapped X mode waves inside the cavitons initially propagate in the
linear X mode on the linear dispersion branch, shrinkage and shortening of
wavelength modifies the linear X mode branch near resonance until it becomes
nonlinear, a rather complicated process which strongly modifies the linear
dispersion. In fact, due to the deviation of the dispersion curve from its
linear topology that is caused by resonant interaction with the energetic
source electron component, the X mode here becomes quasi-electrostatic which
makes it active in affecting the plasma through its ponderomotive force and
particle trapping inside the self-consistently generated cavitons (cf., e.g.,
Schamel, 2022, for the complete nonlinear theory of purely electrostatic
waves). The resonance still occurs beneath though close to $\omega_{ce}$ here
(unless a non-linear frequency shift is taken into account which is assumed of
higher order, here, and thus negligible). Constancy of wave frequency implies
that the nonlinear (about electrostatic) interaction stretches the dispersion
curve out to large wave numbers $k_{nl}$ which substantially exceed the linear
wavenumber $k$, while in frequency remaining below the electron cyclotron
frequency thereby avoiding crossing it upward to approach the upper hybrid
resonance $\omega_{uh}$ which is not allowed by the resonance condition (as
long as it remains linear by itself).

This increase in $k\to k_{nl}$ causes a severe additional reduction of the
phase velocity
$v_{\mathit{ph,nl}}=\omega/k_{nl}\propto\omega/\sqrt{|E_{X}|^{2}}$. In
addition the sudden self-consistent trapping of the waves inside the cavitons
(resulting from the ponderomotive force) implies a violent retardation of the
group speed to $v_{g}\sim c_{ia}$ which adjusts it to the ion acoustic
velocity. The slow down in phase velocity substantially increases the spatial
amplification rate, as has been argued above. As consequence, the waves
locally experience many e-foldings and consequently also large wave
amplification due to increase of their e-folding length. As a result of these
two effects, the amplitude of the trapped X mode waves will readily grow to
become large, which on the one hand causes additional deepening of the
cavitons, while also strongly supporting wave-wave interaction and second
harmonic radiation $\omega_{X}\lesssim 2\omega_{ce}$. Higher harmonic
resonance, as explained above, has little chance to cause comparably intense
radiation. Resonance $x=\omega/\omega_{ce}\lesssim 1$ at the lower X mode
branch in contrast amplifies the confined short-wave lengths X modes (picked
up, for instance, from thermal electromagnetic background noise, cf., e.g.,
Yoon & Lopéz, 2017) which propagate at strongly reduced phase velocity and
experience many e-foldings.

##  3 Hyperbolic resonance

The nonlinear deformation of the linear dispersion has another profound effect
on the topology of resonance that so far (at least to our knowledge) has been
missed in the literature on the ECMI. This is uncovered when considering the
resonance condition (4). The linear-state relativistic resonance is the famous
ellipse in $\vec{\beta}$ or $\vec{p}$-space. In the nonlinear state, however,
there arises the possibility for the ellipse to turn into a hyperbola

$\frac{(p_{\|}-p_{0})^{2}}{a_{\|p}^{2}}-\frac{p_{\perp}^{2}}{a_{\perp
p}^{2}}=1$ (34)

This happens when from Eq. (5) during nonlinear evolution and increasing
wavenumber $\kappa=kc/\omega_{ce}$ the conserved normalized frequency
$x=\omega/\omega_{ce}<\kappa_{\|}$ for some oblique propagation angle $\theta$
drops below the parallel normalized nonlinear wave number $\kappa_{\|}$. (One
may note that this leaves the numerators in (5) invariant such that only the
transverse radius $a_{\perp p}$ is affected to become imaginary). The
condition includes only the parallel wave number and is thus simply a
condition on the the angle $\theta$ of propagation which means that

$\cos\theta>\omega/ck_{nl}=\frac{v_{\mathit{ph,nl}}}{c}$ (35)

exceeds the ratio of the strongly decreased non-linear phase velocity
$v_{\mathit{ph,nl}}$ to light speed. It says that for the topological switch
to take place the angle of resonance should turn more parallel. This condition
is readily satisfied close to the gyrofrequency $\omega_{ce}$ already in the
linear regime, as indicated in Figure 1 where the dispersion begins to flatten
out shortly before crossing the line $x=1$. It essentially excludes
perpendicular propagation from hyperbolic resonance, which however is clear
anyway. The restriction of the resonance to $x<1$ in onsetting nonlinear
evolution and the following smooth increase in wavenumber $k\to k_{nl}$
warrants that this condition is always met at oblique propagation satisfying
the above condition which relaxes in further nonlinear evolution. One may note
that this important change in the resonance topology occurs when working in
the fully relativistic resonance regime which applications have so far
avoided. It is of particular importance in the astrophysically interesting
ultra-relativistic regime.

Hence transition from linear elliptic to nonlinear hyperbolic resonance is
quite natural. In its course the resonating wave becomes increasingly oblique
during nonlinear interaction. On the other hand, it is clear that this case
cannot be realized for any of the resonances $n>1$ which are not confined and
thus do not interact nonlinearly.

Manipulation of the relativistic resonance condition
$x\gamma(\vec{p})=1+\kappa_{\|}p_{\|}/m_{e}c$ with constant $x\approx 1$
yields an upper limit on

$\cos\theta<\frac{\gamma-1}{\gamma\beta_{\|}}\frac{\omega_{ce}}{k_{nl}c}=\frac{\epsilon_{\mathit{kin}}}{m_{e}c^{2}\beta_{\|}}\frac{\omega_{ce}}{\omega_{e}}(k_{nl}\lambda_{e})^{-1}$
(36)

where $k_{nl}$ is the increased nonlinear wavenumber. This is in fact no
serious restriction, as long as the right-hand side exceeds unity. Otherwise
it just excludes a range of nearly parallel propagation angles hence
permitting hyperbolic resonance over a wide oblique angular interval
$\Delta\theta$.

###  3.1 Upward current region

As it turns out, the upward current region becomes the ideal place for
application of the hyperbolic resonance. We shall demonstrate that it is best
suited to fit the celebrated loss-cone distribution as primary source for
generation of AKR here.

Observations in the presumable upward-current AKR source region of the auroral
magnetosphere (e.g., Pottelette & Treumann, 2005) suggest that the kinetic
energy of source electrons is about $\epsilon_{\mathit{kin}}\sim 10$ keV,
while the frequency ratio amounts to about $\omega_{ce}/\omega_{e}\sim 10$.
Parallel speeds of the auroral electrons have been measured roughly around
$\beta_{\|}\sim 10^{-1}$. Moreover, the electron inertial length in the
diluted upward current region is of order $\lambda_{e}\sim 0.5$ km. This
yields within the uncertainty of these numbers

$\omega/ck_{nl}<\cos\theta<1/k_{nl}\lambda_{e}$ (37)

As the left-hand side readily holds close to $x=1$, there are no restrictions
on the propagation angle here for any nonlinear wavelengths
$\lambda_{nl}\gtrsim\lambda_{e}$ longer than the electron inertial length, a
rather weak condition only. If $k_{nl}$ would further increase beyond this
limit, the hyperbolic resonance becomes more oblique though still covering a
large angular interval.

It thus seems natural that the resonance in the upward current region on the
lower X-mode branch, the presumably most important domain of its validity,
readily switches from elliptic to hyperbolic which naturally fits the loss-
cone distribution and the interior of the loss-cone well if not much better
than the resonant ellipse.

This is shown in Figure 3 on one example of available highest-resolution FAST
observations of downward electron fluxes in the upward current region,
performed two decades ago (Delory et al., 1998). The figure suggests that, in
contrast to the usually used resonance ellipse, the hyperbolic resonance
conveniently covers the entire inner perpendicular electron flux (velocity
space distribution) gradient $\partial f_{e}/\partial\beta_{\perp}>0$ offered
by the interior of the loss cone for excitation of the ECMI lower-branch
X-mode. This has some clarifying implications on the choice of the source-
electron distribution function. Hyperbolic resonance is clearly in favour just
of the celebrated loss-cone distribution as the main electronic phase-space
source-distribution here where it is continuously observed and theoretically
supported.

The reasonable conclusion is that in the upward current region loss-cone
distributions, like those in Figure 3 (Delory et al., 1998), can under all
circumstances drive the lower branch ECMI just beneath
$\omega\lesssim\omega_{ce}$ (see Figure 1) increasingly unstable and entering
its state of nonlinear evolution towards large wavenumbers $k\lambda_{e}<1$.
This is expected to happen already for rather moderate amplitudes readily
deforming the linear dispersion relation into its nonlinear cousin to
participate in hyperbolic rather than elliptic resonance.

In the dilute upward-current low-pressure plasma background, it should be
stressed that, even rather moderate confined ECMI-X mode amplitudes suffice to
enter the nonlinear state by choosing a suitable initial wavenumber and
frequency interval that the thermal electromagnetic background noise offers
for amplification (for the presence of such noise, which for oblique
propagation exists in particular below and close to $\omega_{ce}$ see, Yoon &
Lopéz, 2017).

Such a switch in resonance suites the upward loss-cone distribution quite well
for a spectrum $\Delta k_{\mathit{res}}$ of wavenumbers $k$ and excitation of
a narrow spectral band $\Delta\omega$ below $\omega_{ce}$. These intervals are
determined by the angular width of the loss cone and the steepness of the
resonant perpendicular momentum-space gradient.

As described in the caption of Figure 3, the circles separate the loss cone
from the unaffected main plasma distribution at low electron speeds. The
observations do not allow for unambiguous identification of a parallel shift
$p_{0}$ of the distribution. The two drawn black hyperbolic lines map the
boundaries of the loss-cone in their upward directed parts while being
artefacts on the downhill side. Tentatively one resonant hyperbola (red) has
been drawn along the boundary of the loss-cone. The entire positive
perpendicular momentum/velocity gradient region in the loss-cone could be
filled with a continuum of such resonant hyperboles. At any fixed resonant
parallel momentum $p_{\|}$ respectively velocity $\beta_{\|}$ this determines
the growth rate $\Gamma_{n=1}(p_{\|},\omega)$ as function of resonant
frequency and provides an estimate of the bandwidth of excitation which
observations in situ the AKR source suggest to be of the order of just few kHz
in so-called elementary events (cf. the discussion in Baumjohann & Treumann,
2022). Unfortunately, this is inhibited by the large scatter of data in the
iso-flux lines.

Radiation in the fundamental X-mode is confined and cannot escape, at least
not locally, unless it becomes scattered and propagates up along though
oblique to the magnetic field until becoming gradually transformed to find
itself on the free-space X-R mode branch in the magnetospheric tail. Otherwise
the large-amplitude X mode remains trapped, as described above, it undergoes
nonlinear wave-wave interaction to generate escaping second harmonic
radiation. Presumably this is the reason for observation of locally intense
though (because of their proximity to $x=1$) very narrow-band plasma-confined
ECMI emissions near the fundamental $x\sim 1$ (Pottelette & Treumann, 2005;
Baumjohann & Treumann, 2022) in the upward current region. Wave-wave
interaction of those confined large amplitude amplified waves transforms the
confined X mode into the free-space escaping X-mode radiation at frequency
$\omega\lesssim 2\omega_{ce}$ below the second harmonic.

###  3.2 Downward current region

The same hyperbolic resonance condition (36) holds of course also in the
downward current region, but the mostly confirmed and reasonable absence of
any loss-cone distribution here demands a different mechanism to cause
positive perpendicular phase space gradients on the source-electron
distribution function along the resonance hyperbole. Otherwise, if no
appropriate electron distribution becomes available in the downward current
region favourable for hyperbolic resonance, one requires $v_{ph}\sim kc$, in
which case the resonance returns to elliptic. It seems that this applies
indeed to the downward current region even though this condition is difficult
to satisfy. It implies that the lower-branch X-mode waves maintain comparably
large phase velocities and the resonance is replaced away from the electron
cyclotron frequency, a condition difficult to achieve.

Unfortunately no comparably high-resolution measurements of the angular
electron fluxes and distribution functions are yet (at least to our knowledge)
available here. The precise form of the electron distributions is thus not
known and would indeed be worth to be focused on in (hopefully planned and
available) future auroral magnetospheric space missions in the spatially
rather narrowly extended auroral downward current regions of which during
substorms there are many in a row. Electron fluxes are upward, low energy,
spatially highly structured and highly variable in time. The corresponding
upward currents are not smoothly distributed over a wide latitudinal interval
as in the upward current region where the comparable smoothness of the upward
sheet current is reflected in the smooth spatial course of the current-
transverse magnetic field component that is typical for a broad about
unstructured main-field-aligned sheet current.

In contrast to the upward current region, the downward main-field-aligned
currents flow in latitudinally-narrow parallel current sheets or current
braids which are kept apart by the comparably very strong external geomagnetic
field. Accordingly the current-transverse magnetic field component fluctuates
considerably spatially, typical for the presence of many narrow current
filaments. The main geomagnetic field is strong enough to compensate the
Lorentz force of these quasi-stationary narrow field-aligned sheet-currents
which attract each other but cannot merge. Located at the boundary of the
auroral cavity, average densities in the downward current region are
comparably high though still in the range $\omega_{e}/\omega_{ce}<1$ (Temerin
& Carlson, 1998). Energies of the upgoing current-carrying current-closing
electrons of ionospheric origin are just around $\epsilon_{\mathit{kin}}\sim$
few keV at most, roughly one order of magnitude less than in the upward
current region. Related upward velocities $\beta_{\|}$ decrease by a factor of
roughly ten, about compensating the decrease in kinetic energy.

The downward currents are intense enough for causing current instability,
including reconnection in strong current-parallel guide fields and, in
particular, nonlinear evolution (Carlson et al., 1998) which structures and
deforms the electron distribution function and generates large numbers of (so-
called) Debye-scale electrostatic structures (Bernstein et al., 1957; Ergun et
al., 1998), ion and electron holes that propagate along the magnetic field.
Debye lengths are the order of $\lambda_{D}\lesssim$ 10 m here, depending on
the exact spatially variable density and temperature. Whether this is in
favour or not of the ECMI remains unclarified. Absence of loss-cones makes the
hyperbolic resonance less attractive requiring rather particular source-
electron phase-space distributions. These may be provided by those nonlinear
Debye-scale structures, current-driven electron holes. It has been suggested
(Muschietti et al., 1999, and others) that holes cause violent deformations of
the electron distribution function digging phase-space holes into it which
contain a highly diluted temperate low pressure electron population and are
bound by denser cold walls in phase space of much higher hole-field
accelerated speed. At these boundaries the phase-space distribution develops
steep gradients which may serve the needs for exciting the ECMI. This model
has been used to propose the action of the ECMI here as well, based on
elliptic, not hyperbolic resonance (Treumann et al., 2011, 2012) in those
incomplete hollow-electron phase space distributions, sometimes called horse-
shoes.

Observed electron holes extend several $\lambda_{D}$ along the magnetic field
but are much less restricted in perpendicular extension, which is limited by
the width of the unstable field-aligned current filaments and electron
gyroradii, the latter being the order of
$r_{ce}=\beta_{\perp}c/\omega_{ce}\lesssim 1$ km for the dominant source
electrons. Trapping of linearly excited X modes in these (for the purpose of
the ECMI pre-existing) plasma depletions (which result in the nonlinear
evolution of current instability, for the most recent complete theory cf.
Schamel, 2022) is preferably at $k_{\perp}>k_{\|}$, i.e. short perpendicular
wavelengths and slightly oblique propagation $\theta\lesssim 90^{\circ}$. This
is supported close to $x=\omega_{X}/\omega_{ce}\sim 1$ on the lower-branch X
mode where wave numbers increase and wave group speeds become low, holding
however only for the perpendicular wavenumber. This means that $\kappa_{\|}<x$
is a reasonable option inside those holes, whose Debye-structured property
applies only to the parallel direction. Consequently, the electron-hole
trapping-hypothesis identifies the resonance with an ellipse instead of the
hyperbole in the adjacent upward current region.

The range of X mode frequencies fits these perpendicular wavelengths which
become trapped in parallel Debye-scale holes. The elliptic resonance
condition, applied to the interior of such oblate holes, supports excitation
of the ECMI. In contrast to the upward current region, ECMI is caused at the
perpendicular boundary of the phase space holes, not the general form of the
meso-scale electron distribution as this, here, is not of the loss-cone
family. Rather it is self-consistently provided by the self-consistent
nonlinear evolution of the electron holes. Generation of electron holes and
related phase space distributions of the hollow/horseshoe type is primarily
independent on the excitation of the X mode, being provided by the nonlinear
evolution of sufficiently strong field-aligned electron currents (as for
electrostatic waves has extensively been reviewed by Schamel, 2022). It is the
particular electron phase space distribution resulting from this nonlinear
evolution which, in the downstream current region, may encourage the
excitation of lower-branch X modes in elliptical or hyperbolic resonance and
support their increase to reach large local amplitudes while also experience
wave-wave interaction to escape into free space.

The lower-branch hole-trapped X mode has large parallel wavelengths
$\lambda_{\|}\gg\lambda_{D}$ along the magnetic field, which by far exceeds
the Debye-scale. It overlaps a large number of Debye-scale low density holes
which are densely chained along the magnetic field effectively experiencing an
amplification from all those holes along the field over its parallel
wavelength. Simultaneously the wave bouncing back and forth synchronously
inside all of them in perpendicular direction. Even though each hole
contributes differently to the growth of the wave, the concerted action of all
holes in the average should lead to a substantial amplification of the X mode.
If this is the case one expects that observations in situ will detect highly
temporarily structured narrow-band and intense confined short perpendicular
wavelength X mode radiation which propagates at group velocity substantially
below light speed. Its frequency will be very close to the local electron
cyclotron frequency $\omega\sim\omega_{ce}$. Growing to large amplitude this
slow radiation will again undergo wave-wave interaction to generate second
harmonic radiation which escapes from the source and can be observed from
remote. The trapped large amplitude fundamental narrow-band X-mode radiation
can, however, be observed only in situ (as for instance in Figure 2 of
Baumjohann & Treumann, 2022) where it moves together with the entire chain of
holes along the magnetic field. If remaining trapped for all the life-time of
the holes, it will (as has surprisingly been observed, see LaBelle et al.,
2022, and references therein) also be transported away from the source by the
holes either down to the ionosphere (Treumann et al., 2012a) or upward into
the magnetosphere. Similar observation in the upper ionosphere (Parrot &
Berthelier, 2012; Parrot et al., 2022) have also been reported from DEMETER
spacecraft observations.

##  4 Conclusions

Examination of the fully relativistic ECMI resonance condition in application
to the auroral magnetosphere reveals a so far missed domain where the elliptic
resonance turns into a hyperbolic resonance therefore becoming topologically
completely different matching the observation of loss-cones in the auroral
upward current region and suggesting a number of interesting facts. Firstly,
loss cone distributions with harrow and partially filled loss cones can
nevertheless in fully relativistic theory become efficient sources of rather
narrow band excnitation of lower branch X modes which cannot escape from the
source region. The bandwidth in frequency of those modes would be determined
by the narrow loss cone boundary at given wave number which by observation is
of the order of about at most few kHz. It can be observed only in situ the
ECMI source. Second, these locally excited waves have low phase and group
velocities, are trapped in the source region, experience several
exponentiations, and evolve up to large amplitudes. This generates
nonlinearities and causes the wave numbers to shrink further until the
nonlinear wave to become trapped in cavities where the wave undergoes wave-
wave interaction to generate second harmonic radiation above the upper X-mode
cut-off on the free space mode. From there the radiation escapes. This picture
rounds up the theory of AKR excitation by the ECMI in the very low density
upward current region.

Whether the ECMI can also work in the downward current region, remains
uncertain. Primarily the condition of very low density is not necessarily
given there. In that case the ECMI should not evolve. However, the downward
current region is the location of very many so-called Debye-scale structures,
very low density electron holes which along the field are of length of tens of
Debye lengths but in perpendicular direction extend over many electron gyro-
radii corresponding to the transverse wavelength of the X mode. In
perpendicular direction the X mode fits the hole and, in the low density
region of the holes, becomes as well amplified and evolves nonlinearly. ECMI
generated AKR could thus also result from here though in completely different
ways.

In addition, however, in the downward current region Z mode radiation may
compete in all those regions which do not experience dilution by Debye
structures. Here the Z mode propagates and is excited by VLF, undergoes wave
wave interaction and radiates at second and possibly even higher harmonics.
This mechanism is not based on the ECMI but deserves to be taken into account
when dealing with the downward current region.

Another most interesting observation of AKR is that it seems to weakly leak
down into the ionosphere at spacecraft altitudes and even down to the ground
under favourable though still unknown conditions. These observations pose the
interesting question: How can AKR generated at altitudes of at least 1000 km
above ground pass into and even across the ionospheric density screen? Such a
passage seems forbidden. However, if generated in density cavities, either
self-generated cavities in the upward or within Debye scale cavities in the
downward current region, then X modes could possibly be trapped inside those
cavities and together with them may become transported down to the ionosphere
where few of them could survive the transport and would leak out. This most
interesting observation still awaits its theoretical explanation.

This work was part of a brief Visiting Scientist Programme (of RT) at the
International Space Science Institute Bern. We acknowledge the interest of the
ISSI directorate as well as the hospitality of the ISSI staff, in particular
the assistance of the librarians Andrea Fischer and Irmela Schweitzer, and the
Systems Administrator Willi Wäfler.

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