# Equatorial source of oblique electromagnetic ion cyclotron waves:
peculiarities in the ion distribution function

###### Abstract

Electromagnetic ion cyclotron (EMIC) waves are important for Earth’s inner
magnetosphere as they can effectively drive relativistic electron losses to
the atmosphere and energetic (ring current) ion scattering and isotropization.
EMIC waves are generated by transversely anisotropic ion populations around
the equatorial source region, and for typical magnetospheric conditions this
almost always produces field-aligned waves. For many specific occasions,
however, oblique EMIC waves are observed, and such obliquity has been commonly
attributed to the wave off-equatorial propagation in curved dipole magnetic
fields. In this study, we report that very oblique EMIC waves can be directly
generated at the equatorial source region. Using THEMIS spacecraft
observations at the dawn flank, we show that such oblique wave generation is
possible in the presence of a field-aligned thermal ion population, likely of
ionospheric origin, which can reduce Landau damping of oblique EMIC waves and
cyclotron generation of field-aligned waves. This generation mechanism
underlines the importance of magnetosphere-ionosphere coupling processes in
controlling wave characteristics in the inner magnetosphere.

JGR: Space Physics

Department of Physics, University of Texas at Dallas, Richardson, TX, USA
Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles, USA

David S<EMAIL_ADDRESS>
We report observations of very oblique EMIC waves around their equatorial
source region

Oblique EMIC wave generation is associated with field-aligned thermal ion
streams and hot, transversely anisotropic ions

The presence of field-aligned thermal ion streams attributes oblique EMIC wave
generation to its possible ionospheric source

##  1 Introduction

Electromagnetic ion cyclotron (EMIC) waves is a natural emission generated by
transversely anisotropic ion (proton) populations in the Earth’s inner
magnetosphere [Cornwall . (1970), Cornwall Schulz (1971), Min . (2015), Yue .
(2019)]. These waves are responsible for resonant scattering of relativistic
electrons and energetic (ring current) ions [<]see review¿[and references
therein]Usanova&Mann16, which makes this wave mode principally important for
the inner magnetosphere dynamics.

Transversely anisotropic ion populations, either injected from the plasma
sheet or formed due to the magnetosphere compression by solar wind transients
[Jun . (2019), Jun . (2021), Kim . (2021)], usually generate field-aligned
EMIC waves [Chen . (2010), Chen . (2011)], whereas generation of oblique waves
is often suppressed by wave Landau damping due to suprathermal ions [<]see
discussions in¿SoriaSantacruz13. Thus, oblique EMIC waves have been mostly
detected off the equator [Liu . (2013)], because wave propagation in highly
inhomogeneous magnetic field and plasma will result in wavevector deviation
from the field-aligned direction [Rauch Roux (1982), Thorne Horne (1997),
Fraser Nguyen (2001)]. It has been shown that oblique EMIC waves can scatter
particles via several resonant mechanisms not available to field-aligned waves
[<]see review by¿Usanova21. First, wave obliquity modifies the efficiency of
relativistic electron scattering
[<]see¿Khazanov&Gamayunov07,Lee18:emic,Hanzelka23. Second, the Landau
resonance between oblique EMIC waves and cold ions allows EMIC waves to serve
as a transmitter of energy between hot and cold ion populations [Omura .
(1985), Kitamura . (2018), Ma . (2019)]. Third, oblique EMIC waves can
resonate with cold plasmasphere electrons [<]in linear and nonlinear regimes,
see¿Wang19:emic and can accelerate and/or precipitate them to form the stable
red aurora arcs [Cornwall . (1971), Thorne Horne (1992)]. Fourth, oblique EMIC
waves may scatter energetic ($\sim 100$keV) electrons via the bounce resonance
[Q. Wang . (2018), Blum . (2019)]. All these mechanisms imply the importance
of understanding possible sources of oblique EMIC waves: can such waves be
generated around the equator or only off-equator as the field-aligned waves
propagate away from their equatorial source?

The equatorial generation of oblique EMIC waves requires a viable mechanism to
suppress the Landau damping, i.e., reduction of the field-aligned gradient in
the ion phase space density around the Landau resonant energies (hundreds of
eV). For the electron-scale whistler-mode waves, it has been shown that
oblique wave generation can be explained by field-aligned electron streams
generated by ionosphere outflow [<]see discussions
in¿[]Artemyev&Mourenas20:jgr, which largely suppress Landau damping [<]see,
e.g.,¿Mourenas15,Li16. Such ionosphere outflows also include field-aligned
ionospheric ion populations, which are often detected in the inner
magnetosphere and near-Earth plasma sheet [Yue . (2017), Artemyev . (2018)].
Therefore, oblique EMIC wave generation may be explained by a combination of
near-equatorial, transversely anisotropic hot ions and field-aligned thermal
ion streams.

In this study, we investigate near-equatorial observations of very oblique
EMIC waves by Time History of Events and Macroscale Interactions during
Substorms (THEMIS) spacecraft [Angelopoulos (2008)]. Combining measurements of
ion distribution functions [McFadden . (2008)] and linear dispersion solver
[Astfalk Jenko (2017)], we reveal the conditions of such wave generation and
demonstrate that very oblique EMIC waves are associated with field-aligned ion
streams. The rest of this paper is structured as follows: Section 2 describes
THEMIS instrumentation, plasma and wave measurements, methods of data
analysis, and details of the linear dispersion solver. Section 3 examines the
generation of multiple very oblique EMIC wave events by analyzing the observed
ion distribution function and comparing this distribution with that during
field-aligned EMIC waves. Section 4 summarizes our results and discusses their
implication for modelling of particle dynamics in the inner magnetosphere.

##  2 Dataset and instruments

We combine near-equatorial THEMIS measurements of EMIC waves [<]of 16Hz
sampling rate from the flux-gate magnetometer, see¿Auster08:THEMIS, plasma
sheet ion distributions [<]at 3s resolution from the electrostatic analyzer
(ESA), covering $<25$keV energy range and full pitch-angle range,
see¿McFadden08:THEMIS, total plasma density inferred from the spacecraft
potential [<]see¿Bonnell08,Nishimura13:density, and the numerical dispersion
solver for electromagnetic waves – Linear Electromagnetic Oscillations in
Plasmas with Arbitrary Rotationally-symmetric Distributions [<]LEOPARD,
see¿Astfalk&Jenko17. We use the single-spacecraft maximum variance analysis
technique [Means (1972)] to estimate the wave normal angle, which comes with
an ambiguity of parallel versus anti-parallel directions: the wave normal
angle from this technique ranges from $0^{\circ}$ to $90^{\circ}$, rather than
$0^{\circ}$ to $180^{\circ}$.

An important advantage of LEOPARD model is it can accommodate arbitrary
gyrotropic distribution functions in a uniform grid of velocities parallel and
perpendicular to the background magnetic field, $(v_{\parallel},v_{\perp})$.
Compared to dispersion solvers that require fitting the measurements to
prescribed particle distributions (such as Maxwellians), LEOPARD significantly
reduces the uncertainties due to fitting and hence can give a more reliable
dispersion solution. Therefore, we use ESA ion distributions in $23$
logarithmically spaced energy channels between $5$ and $25$ keV and $8$
linearly spaced pitch-angle channels and interpolate them to a denser
$(v_{\parallel},v_{\perp})$ grid, which is then passed to LEOPARD to evaluate
the observed EMIC wave dispersion and growth rate. We will combine LEOPARD and
THEMIS measurements to reveal properties of specific ion distributions that
are responsible for the generation of very oblique EMIC waves, quite atypical
type of EMIC emission. Note that although THEMIS ESA provides plasma sheet
($<30$keV) electron distributions, this electron population usually does not
resonate with EMIC waves or alter the wave dispersion. Thus, we only treat
electrons as the cold background plasma (with electron $\beta=10^{-2}$) and do
not examine their contribution to EMIC wave growth.

##  3 Oblique EMIC events

We now analyze in details multiple events with very oblique EMIC waves. These
are typical events of this wave population and hence their characteristics
will be representative of the entire population.

###  3.1 Detailed analysis of the first event

The first event shows very oblique EMIC waves observed by THEMIS-A spacecraft
on February 17, 2020, between 17:50-17:58 UT (Figure 1). During this event,
hydrogen-band EMIC waves (of frequencies within $0.3-0.9$ of the proton
cyclotron frequency $f_{cH+}$) were detected in the dawn flank of the outer
edge of the inner magnetosphere (MLT $\sim 07$, $L\sim 10$) with the presence
of hot, plasma sheet ions. The EMIC wave normal angle can reach $80^{\circ}$
during this event. The ion beta is $\beta\sim 1.7$, typical for the inner
plasma sheet edge/outer edge of the inner magnetosphere [Yue . (2017),
Artemyev . (2018)].

Figure 1 shows the field-aligned population of thermal ($\lesssim 1$ keV) ions
and hot ($>2$keV), transversely anisotropic ion population. Both the flux and
anisotropy of hot ions are large right around the moment of intense, very
oblique EMIC waves. Such transversely anisotropic ions are likely responsible
for EMIC wave generation [Chen . (2010), Chen . (2011), Yue . (2019)], but
instead of more typical field-aligned waves we observe very oblique waves.
Therefore, certain features in the ion distribution significantly alter the
generation mechanism. Most likely the thermal field-aligned ion population
affects wave generation and moves the positive growth rate to high wave normal
angles. To verify this assumption, we will combine the measured ion
distribution and linear dispersion solver.

Figure 1: Observation of oblique, hydrogen-band EMIC waves by THEMIS-A
spacecraft on February 17, 2020. a) Wave magnetic field power spectrum, b)
wave normal angle; top and bottom lines represent hydrogen ion $H^{+}$ and
helium ion $He^{+}$ cyclotron frequencies, respectively; vertical dashed lines
mark the time interval that is used to average the ion phase space density for
subsequent investigations of wave dispersion properties. c-g) Ion pitch-angle
distributions for different energy ranges as a function of time.

Both the field-aligned thermal ion population and hot, transversely
anisotropic population can be well seen in Figure 2, where we plot the 1-min
averaged (around the time of the most intense wave spectrum) ion distribution
in $(v_{\parallel},v_{\perp})$ plane. The velocities are normalized to the
Alfv’en velocity $v_{A}=B/\sqrt{4\pi nm_{p}}$, where $n$ is the plasma density
calculated from spacecraft potential, $m_{p}$ is proton mass. Compared with
the isotropic distribution (shown in black dashed curves), observations
clearly show a strong transverse anisotropy at $v_{\perp}>0.5v_{A}$ and field-
aligned anisotropy at $v_{\perp}<0.25v_{A}$. This $(v_{\parallel},v_{\perp})$
distribution is then passed to the LEOPARD solver [Astfalk Jenko (2017)] to
calculate the wave dispersion relation, where we can determine the
frequency–wave normal angle region of positive growth rate and compare this
with observations of very oblique EMIC waves. Figure 3 shows results of
LEOPARD calculations: the positive growth rate is indeed confined to very
oblique wave normal angles, peaking at $74^{\circ}$, and to wave frequencies
around $0.35-0.55f_{cH+}$ (note throughout the paper we also use
$\Omega_{ci}=2\pi f_{cH+}$). These ranges of wave normal angles and
frequencies are quite close to those observed by THEMIS (see Fig.1), which
confirms that the ion distribution from Fig. 2 can reliably reproduce the main
wave properties. Note that there is no positive growth rate for field-aligned
(or small wave normal angle) waves (not shown).

Figure 2: Snapshot of the ion distribution observed by THEMIS-A during the
event on February 17, 2020 from Fig. 1; ESA measurements are averaged over
17:53:30-17:54:30 UT, which are used to plot the contours of the ion phase
space density (blue). For reference, the black dashed traces show contours of
the particle energy. Figure 3: The estimated EMIC wave growth rate in the
plane of wave normal angle and wave frequency. The growth rate is evaluated
for hydrogen-electron plasma with the ion distribution from Fig. 2 .

To reveal the ion distribution that is responsible for the generation of
oblique EMIC waves, we decompose the observed $(v_{\parallel},v_{\perp})$
distribution into three ion populations, each fitted by a bi-maxwellian
distribution. Figure 4 shows (a) the observed distribution, (b) the cold ion
population with $\beta\sim 10^{-2}$ (this population does not contribute to
EMIC wave generation, but is needed to realistically match the total ion
density with observations), (c) sum of cold and thermal, with
$\beta_{\perp}/\beta_{\parallel}\sim 0.45$, ion populations (the latter one is
parallel anisotropic and used to represent the $<1$keV field-aligned ion
streams), (d) sum of cold, thermal, and hot, with
$\beta_{\perp}/\beta_{\parallel}\sim 3.8$, ion populations (the latter one is
transversely anisotropic and provides free energy for EMIC generation).
Comparison of panels (a) and (d) in Fig. 4 shows that this three-component
fitting agrees well with the observed ion distribution. Using this fitted
distribution, we then evaluate the contribution of each ion population to the
EMIC wave dispersion and generation. Figure 5 shows the dispersion relations
of EMIC waves with $74^{\circ}$ propagation angle from the observed ion
distribution and from the fitted distributions in Figs. 4 (b-d). For
comparison, we also show the cold plasma dispersion for field-aligned waves
[Stix (1962)]. Solutions of the wave dispersion for cold-only (blue trace) and
cold $+$ thermal (cyan trace) populations are quite similar, apart from the
stronger damping in the latter. Introducing the hot transversely anisotropic
population (orange trace) results in positive growth rate, which is larger
than the growth rate from the observed ion distribution, but occupies a
similar frequency range. The main role of the thermal, field-aligned
anisotropic population is to provide strong cyclotron damping of low wave
normal angle waves: note that the energy of the first cyclotron resonance
decreases with wave normal angle decrease and reaches $1$keV (around the
energy of the thermal population) for field-aligned waves. A secondary, yet
important, role of the thermal population is in reducing the total ion
anisotropy: although the anisotropy of the entire ion distribution,
$\beta_{\perp}/\beta_{\parallel}=1.8$, does not exceed much the threshold for
field-aligned EMIC generation [Yue . (2019)], the anisotropy of the hot
population is sufficiently large ($\beta_{\perp}/\beta_{\parallel}\approx 4$)
to produce very oblique waves [<]see the analogical mechanism of oblique
whistler-mode wave generation by the highly anisotropic electron component
in¿Gary11:pop

Figure 4: Observed ion distribution during the event on February 17, 2020 (a)
and its fitting to a sum of three bi-maxwellian distributions (b-d): panel (b)
shows a single cold ion population with $\beta<10^{-2}$, panel (c) shows a
summation of cold and thermal ion populations, panel (d) shows a summation of
cold, thermal, and hot ion populations. Figure 5: Solutions of the wave
dispersion (left) and wave growth rate (right) for EMIC waves with
$74^{\circ}$ wave normal angle: results for the observed ion distribution are
shown in red, results for the three bi-maxwellian distributions from Figure 4
are shown in blue (cold population only), cyan (cold and thermal populations),
and orange (cold, thermal, and hot populations), respectively. Left panel
shows wave dispersion relation: $\Omega_{ci}$ is the proton cyclotron
frequency, $d_{i}=c/\omega_{pi}$ is proton inertial length. Black line shows
the analytical solution for field-aligned EMIC waves in cold proton-electron
plasma with background parameters from observations during the event [Stix
(1962)]. Black dashed traces in the right panel show the absolute value of the
growth rate, $|\gamma|=\omega$.

We also investigate the contribution of different resonances to the oblique
EMIC wave dispersion and growth rate. Figure 6 shows the result for the first
cyclotron resonance only, $n=1$ (orange), first cyclotron and Landau
resonances $|n|\leq 1$ (cyan), and for all resonances up to $|n|=10$ (red).
There is barely any difference between wave dispersions for these three cases,
i.e., the wave dispersion is mainly provided by the ion population
contributing to the first cyclotron resonance. The comparison of wave growth
rates shows that the same ion population is responsible for wave generation,
whereas Landau damping reduces the magnitude of the growth rate, especially at
small wave frequencies.

Figure 7 further confirms the principal role of the first cyclotron resonance
in generating the observed oblique EMIC waves, by combining contours of
constant phase phase density (from Fig. 2) and resonant conditions for two
approximations. The grey shaded region in the left panel shows the cyclotron
resonant velocities calculated from the dispersion relation for $74^{\circ}$
wave normal angle and wave frequencies of positive growth rate,
$\omega/\Omega_{ci}\in[0.38,0.55]$. Comparison of constant phase phase density
contours (blue traces) and resonance curves (contours of constant energy in
the wave rest frame, shown in red) demonstrates that within the grey shaded
region the gradient along the resonance curves (seen from the comparison of
constant energy curves, shown in dashed black, and phase space density
contours, shown in blue) corresponds to an increase in the phase space
density, which will drive EMIC wave growth [Lyons Williams (1984)]. Similarly,
the same conclusion can be drawn for the resonant velocity range for
monochromatic waves, with a frequency corresponding to the peak wave intensity
in observations and variations of the resonance condition along magnetic
latitudes (right panel): the positive gradient of the ion phase space density
along resonance curves within the purple shaded region will amplify the waves
as they propagate away from the generation region near the equator.

Figure 6: Solutions of the wave dispersion (left) and wave growth rate (right) for $74^{\circ}$ wave normal angle, while keeping different orders of resonances in the dielectric tensor calculation. The result from the observed distribution for all resonances (in red, same as the red trace in Figure 5) includes up to $|n|=10$ resonances. |
---|---
Figure 7: Contours of constant phase space density from Figure 2 (blue), with
resonant velocities for the first cyclotron resonance (shaded region), and
resonant curves corresponding to contours of constant ion energy in the wave
rest frame (red). Orange vertical line marks the Landau resonant energy. Left
panel shows the range of resonant velocities at the equator for the frequency
range with positive growth rate (grey shaded area), right panel shows the
range of resonant velocities for a fixed equatorial frequency
$0.47\Omega_{ci}$ and a latitudinal range of $0^{\circ}\leq\lambda\leq
25^{\circ}$ (purple shaded region).

###  3.2 Comparison of ion distributions during field-aligned and oblique EMIC
events

Figure 8: Wave magnetic field power spectrum (left column), ion distribution
(center column), and wave growth rate (right column) for oblique (a-e) and
field-aligned (f) EMIC wave events. In the wave power spectra: for oblique
EMIC waves, red contours show the domain where wave normal angle
$>60^{\circ}$, whereas for field-aligned EMIC wave event orange contours show
the domain where wave normal angle $<20^{\circ}$; the two horizontal white
lines mark hydrogen and helium ion gyrofrequencies from top to bottom. During
the February 14, 2020 event (b), part of the wave spectrum has been cut to
remove the spin tones that obscure the waves.

Figure 8(b-e) shows four more examples of THEMIS observed very oblique EMIC
waves in the dawn flank. All these events share similar properties of the
event discussed in Section 3.1:

  * •

Waves are proton band EMIC waves with clear maximum of the wave intensity
around $[0.3,0.5]$ of local proton cyclotron frequency and with wave normal
angles typically exceeding $60^{\circ}$ (left panels).

  * •

Ion distribution functions (middle panels) observed around the oblique EMIC
wave burst include a very anisotropic hot (a few keVs) ion population
($\beta_{\perp}/\beta_{\parallel}>3$) and a field-aligned anisotropic thermal
($\leq 1$keV) ion population. The latter can lead to strong cyclotron damping
of field-aligned EMIC waves.

  * •

Combining the measured ion distribution and the linear dispersion solver,
LEOPARD, we show positive growth rates for very oblique EMIC waves (right
panels), as well as damping for the field-aligned waves (not shown).
Therefore, the most important condition for very oblique EMIC wave generation
is likely the combination of transversely anisotropic hot ions (providing wave
growth via cyclotron resonance) and field-aligned thermal ions (providing
cyclotron damping of small wave normal angle waves).

To verify the importance of the thermal field-aligned population for very
oblique EMIC wave growth, we further selected a typical field-aligned EMIC
wave event observed by THEMIS in the dusk flank. As shown in Figure 8(f), in
the absence of field-aligned thermal ions, the field-aligned waves can be
generated via cyclotron resonance with transversely anisotropic hot ions.
Moreover, in the absence of the field-aligned population, the anisotropy of
the entire ion distribution is mainly determined by the anisotropy of hot
ions, which does not need to be as large as in events with oblique waves in
order to generate field-aligned waves. As a result, in the event from Fig.
8(f) the hot ion anisotropy, $\beta_{\perp}/\beta_{\parallel}=1.3$, is
sufficient to generate field-aligned EMIC waves, but insufficient to generate
oblique waves (the growth rate of oblique waves is negative for this event;
not shown).

##  4 Discussion and conclusions

In this study we analyze several events with THEMIS observations of very
oblique EMIC waves around the equator. During these events, the observed ion
distributions consist of a highly transversely anisotropic
($\beta_{\perp}/\beta_{\parallel}\sim 4$, $>2$keV) hot ion population and a
field-aligned anisotropic thermal ion population
($\beta_{\perp}/\beta_{\parallel}\sim 0.3$, $<1$keV). It is this field-aligned
thermal ion population that prohibits the generation of field-aligned EMIC
waves: the cyclotron resonant energy of waves with small wave normal angles is
$\sim 1$keV, where the transverse anisotropy is insufficient to drive these
waves. Therefore, such thermal ($<1$ keV) field-aligned ion population play a
key role in producing very oblique EMIC waves, which are often observed on the
dawn flank. The energy range and field-aligned anisotropy of this population
suggest that these are likely ionospheric outflow ions [<]see statistics of
ion pitch-angle distributions in¿[]Artemyev18:jgr:RBSP&THEMIS,Yue17:ions.
Overlapping of such outflow, probably enhanced at the dawn flank due to strong
plasma sheet electron precipitation driven by whistler-mode waves [Thorne .
(2010), Ni . (2016)], and the hot ion population, likely drifted from the dusk
flank after being injected from the plasmasheet
[<]e.g.,¿Birn97:ion,Gabrielse14,Ukhorskiy18:DF, creates favorable conditions
for the generation of very oblique EMIC waves. This further implies that very
oblique waves are likely a result of magnetosphere-ionosphere coupling, in
contrast to the more typical field-aligned waves generated in the dusk flank
due to plasmasheet injections or on the day side due to magnetosphere
compression by the solar wind [Yue . (2019), Jun . (2019), Jun . (2021)].

Figure 9: Minimum resonant energies for protons (blue, linear scale) and
electrons (red, logarithmic scale) in Landau resonance and first cyclotron
resonance ($"+1"$ for protons, $"-1"$ for electrons). Dashed and dotted lines
are calculated using generic bi-maxwellian distributions (with
$\beta=10^{-2}$), whereas solid and dot-dashed lines are based on dispersion
results for the first event (Figure 8 a).

In addition to the possible resonant interactions between oblique EMIC waves
and magnetospheric particles, as discussed in the introduction, a new resonant
mechanism has been proposed recently that makes oblique EMIC waves potentially
more important. It has been shown by Hanzelka23:emic that very oblique and
sufficiently intense EMIC waves may resonate with energetic electrons via the
so-called fractional (or subharmonic) resonances [Lewak Chen (1969), Smirnov
Frank-Kamenestkiǐ (1968)]. In contrast to cyclotron resonances with integer
resonant numbers, the factional resonance is a purely nonlinear effect
providing electron scattering in resonances with fractional numbers [Terasawa
Matsukiyo (2012)], which reduces the electron energy in resonance with EMIC
waves to sub-relativistic values [Hanzelka . (20232)]. THEMIS observations of
very oblique EMIC waves and the proposed formation mechanism of these waves
suggest that sub-relativistic electron precipitation on the dawn side [<]see
statistics of such precipitation in¿[]Tsai23 may be partly driven by EMIC
waves, not exclusively by whistler-mode waves.

Figure 9 shows the energies of electrons and ions in resonance with very
oblique EMIC waves (we use a wave normal angle of $74^{\circ}$ and plot
resonant energies as a function of wave frequency for typical plasma
frequencies as from Figs. 8). Note that the electron resonant energies for
fractional resonances fall between the regions bounded by the first cyclotron
resonance, $n=-1$, and Landau resonance, $n=0$ [Hanzelka . (20232)]. Comparing
the results for the cold plasma dispersion ($\beta=10^{-2}$) and for the
observed hot plasma with $\beta\approx 1.7$, we can see a higher resonant
energy for the hot plasma case due to a decrease of the wave number at a fixed
frequency [Silin . (2011)]. Cyclotron and Landau resonant energies for protons
are within $[1,10]$keV for $\omega/\Omega_{cp}<0.5$, where most of wave power
is observed (see Fig. 8 (a)). This energy range allows oblique EMIC waves to
heat thermal ($\sim 1$keV) ions and scatter ring current ($\sim 10$keV) ions
into the loss cone (note that these resonant energies are calculated at the
equator, which will increase in the off-equatorial region). The range of
resonant energies for electrons, on the other hand, is much wider: from $\sim
1$eV in Landau resonance to $1$MeV in cyclotron resonance. The fractional
resonances with a resonance number $\in[0,1]$ will fill this gap and allow
EMIC waves to also scatter hundreds of keV electrons. This is likely the most
interesting and potentially important implication of very oblique waves.

To conclude, using THEMIS observations in the inner magnetosphere, we have
investigated the generation mechanism of very oblique EMIC waves. Six typical
events of such waves are observed at the dawn flank, in contrast to the more
common field-aligned waves observed at the dusk and noon sectors. Very oblique
EMIC waves are usually accompanied by ion distributions consisting of two main
populations (except for the cold plasma population contributing to the total
density). 1) A thermal ion population, at $\leq 1$ keV, with the field-line
anisotropy providing the cyclotron damping of field-aligned EMIC waves and
potentially reducing the Landau damping of oblique EMIC waves. This population
shares the same properties of ionospheric outflow as reported previously for
the inner magnetosphere [<]e.g.,¿Yue17:ions. 2) The hot ion population, at
$>5$keV, with a strong transverse anisotropy
($\beta_{\perp}/\beta_{\parallel}\sim 4$) providing the cyclotron resonant
growth of oblique EMIC waves with wave normal angles exceeding $60^{\circ}$.
These observations underline the importance of magnetosphere-ionosphere
coupling in producing very oblique EMIC waves.

###### Acknowledgements.

We acknowledge the support of NASA contract NAS5-02099 for the use of data
from the THEMIS Mission, specifically K. H.Glassmeier, U. Auster and W.
Baumjohann for the use of FGM data (provided under the lead of the Technical
University of Braunschweig and with financial support through the German
Ministry for Economy and Technology and the German Center for Aviation and
Space (DLR) under contract 50 OC 0302). Work of D.S.T., X.-J. Z., and A.V.A.
are supported by NSF grant #2329897, and NASA grants #80NSSC20K1270,
#80NSSC23K0403, #80NSSC23K0108 .

## Open Research

THEMIS data is available at http://themis.ssl.berkeley.edu. Data access and
processing was performed using SPEDAS V4.1 and its Python-based
implementation, see Angelopoulos19 and [Grimes . (2022)].

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