# Resonant whistler-electron interactions: MMS observations vs. test-particle
simulation

###### Abstract

Simultaneous observation of characteristic 3-dimensional (3D) signatures in
the electron velocity distribution function (VDF) and intense quasi-
monochromatic waves by the Magnetospheric Multiscale (MMS) spacecraft in the
terrestrial magnetosheath are investigated. The intense wave packets are
characterised and modeled analytically as quasi-parallel circularly-polarized
whistler waves and applied to a test-particle simulation in view of gaining
insight into the signature of the wave-particle resonances in velocity space.
Both the Landau and the cyclotron resonances were evidenced in the test-
particle simulations. The location and general shape of the test-particle
signatures do account for the observations, but the finer details, such as the
symmetry of the observed signatures are not matched, indicating either the
limits of the test-particle approach, or a more fundamental physical mechanism
not yet grasped. Finally, it is shown that the energisation of the electrons
in this precise resonance case cannot be diagnosed using the moments of the
distribution function, as done with the classical ${\bf E}.{\bf J}$
“dissipation” estimate.

JGR: Space Physics

Laboratoire de Physique des Plasmas, CNRS - École Polytechnique - CNRS -
Sorbonne Université - Unversité Paris Saclay- Observatoire de Paris-Meudon,
F-91128 Palaiseau Cedex, France LESIA, Observatoire de Paris, PSL Research
University, CNRS, UPMC Université Paris 6, Université Paris-Diderot, Meudon,
France Physics and Astronomy Department, University of Florence, Sesto
Fiorentino, Italy

Etienne<EMAIL_ADDRESS>
Characteristic double-branch signatures in the electron Velocity Distribution
Function (VDF) are observed simultaneously with a whistler wave.

The wave, applied to test-particles, produces signatures in the VDF through
Landau and cyclotron resonances.

This resonant wave-particle interaction cannot be diagnosed in the
Magnetospheric MultiScale (MMS) observations through the dissipative
$\mathbf{E}\cdot\mathbf{J}$ term.

##  1 Introduction

Resonant wave-particle interactions are one of the few mechanisms in
collisionless plasmas that enable a net transfer of energy from oscillating
electromagnetic fields to moving charged particles. They play a fundamental
role in various regions of the near-Earth plasma environment, such as the
bowshock, the radiation belts, the polar cusp or the magneto-tail [Mazelle .
(2000), Grison . (2005), Thorne (2010), Fujimoto . (2011), Krasnoselskikh .
(2013)]. In (fully developed) plasma turbulence, wave-particle interactions
are also thought to play a leading role in dissipating energy as the turbulent
cascade proceeds from large (fluid) to small (kinetic) scales [Bruno Carbone
(2013), Sahraoui . (2020)]. In the solar wind (and to some extent the
magnetosheath), the most debated dissipation processes are the Landau damping
[Landau (1946), Howes . (2008), Schekochihin . (2009), Gary Smith (2009),
Sahraoui . (2010), Podesta . (2010), Sulem Passot (2015), Kobayashi . (2017)],
cyclotron damping [Leamon . (1998), Kasper . (2008), Cranmer (2014), He .
(2015)] and stochastic heating [Chandran . (2010)], which all would involve
different spatial or temporal scales. Often magnetic reconnection is also
evoked as a potential dissipation process in localized current sheets that
self-consistently form in turbulence plasmas [Matthaeus . (1984), Retinò .
(2007), Sundkvist . (2007), Chasapis . (2015)]. However, even within such
localized coherent structures, Landau damping is shown to be very effective in
numerical simulations of collisionless magnetic reconnection [TenBarge Howes
(2013), F. . (2013), Numata Loureiro (2015)].

Despite their role in energy dissipation, a direct diagnosis of wave-particle
resonances in numerical simulations and in-situ data remains elusive. The
difficulty to approach these processes stems from the need to measure
simultaneously the 7D VDF (3 spatial dimensions, 3 velocity dimensions, and
time) with high temporal and velocity space resolutions to access the kinetic
scales, and the 4D structure of the electric and magnetic fields. While the
latter could have been achieved at the magnetohydrodynamic (MHD) and ion
scales using the Cluster data and appropriate data analysis techniques
[Sahraoui . (2006), Narita . (2010), Sahraoui . (2010)], the former became
possible only in recent years thanks to the MMS mission [Burch . (2016)]. MMS
indeed provides us with the highest ever achieved resolution of the particle
VDFs, both in time and velocity space [Pollock . (2016)]. Furthermore, thanks
to its small inter-spacecraft separations ($\sim 10$ km) MMS allows us to
probe in 3D kinetic spatial scales of the fluctuations fields. On the other
hand, the increasing computer capabilities allows achieving Vlasov simulations
with high-enough phase space resolution to unravel the complex nature of the
kinetic dissipation in turbulent collisionless plasmas [Cerri . (2018)]. The
present article is part of the ongoing efforts in this direction.

A few observational approaches on signatures of such mechanisms were
previously proposed, and here we focus on studies manipulating 3-dimensional
VDFs. kitamura2018science reported the unambiguous observation of wave-ion
resonances leading to particle acceleration, associated with an ion cyclotron
wave, in the Earth magnetosphere, displaying clear agyrotropic signatures
(phase bunching) and their time evolution. In the work of gershman2017nc, at
the magnetopause, the energy exchange between electrons and kinetic Alfvén
waves was studied in terms of the dissipative $\mathbf{E}.\mathbf{J}$ term,
and trapped electrons were found in the wave minima. In the radiation belts,
min2014jgr proposed empirical indications for Landau resonance signatures in
the electron VDFs, as local minima in their velocity derivatives close to the
parallel resonant velocity, in the presence of Chorus waves. he2015apj and
marsch2011ag show indications in the solar wind of resonances in the proton
VDFs, composed of a diffused, anisotropic core and secondary beam, which were
linked to kinetic waves. In the Earth magnetosheath and using a field-particle
correlation technique, chen2019nc presented structures in the fluctuating
electron VDF close to the electron therrmal speed, which the authors linked to
electron Landau damping.

The method followed in the present study is similar to the approach of
kitamura2018science and gershman2017nc, in that we first identify a neat
electromagnetic wave, as intense and monochromatic as possible, study its
potential effect on particle Velocity Distribution Functions (VDF), and then
compare the expected signatures with the VDF observed outside and inside the
region where the wave is observed. This approach enables an unequivocal,
3-dimensional comparison of resonant signatures in the VDF between
observations and the simulation.

The wave studied in this work is a high-frequency quasi-parallel whistler
mode, ubiquitous in both magnetospheric and the solar wind plasmas [Tao .
(2012), Lacombe . (2014), Stansby . (2016)]. Electrons are therefore the
species of interest, and the frequencies of both the wave and the electron
motion (gyration) are much higher than the fastest particle instruments
operating in space, though in the reach of wave sensors. Anisotropies in the
electron distribution functions are fundamental for the generation of whistler
waves, as shown in various contexts such as the solar wind [Tong . (2019)],
magnetic reconnection regions [Huang . (2016), Yoo . (2018), Yoo . (2019)], or
mirror mode magnetic holes or other coherent structures in the magnetosheath
[Huang . (2017), Ahmadi . (2018)]. verscharen2019apj, vocks2005apj and
seough2015apj have explored the theoretical link between whistler waves and
solar wind electrons, in either the formation or the scattering of strahl and
halo electrons. In their numerical approach, hsieh2017jgr show how oblique
whistler mode chorus in the magnetosphere can lead to electron acceleration up
to a few MeV, via Landau, cyclotron, and higher order resonances.

To go beyond the 1-dimensional description of wave-particle resonances, in
which a resonance is reduced to its associated (scalar) parallel speed, we
explore the possibilities offered by a test-particle approach for a more
comprehensive description of the mechanism and a direct comparison with the
observations. This approach also presents the great advantage of isolating the
effect of the wave on the particles, with no feedback allowed. A succinct view
on particle energisation is also proposed, in order to appreciate whether the
energy gained by the resonant particles can be quantified in observations.

##  2 Particle data analysis

The particle data used in this study were recorded by the Fast Plasma
Investigation (FPI) of the Magnetospheric Multiscale (MMS) mission [Burch .
(2016), Pollock . (2016)]. We work in a reference frame in which the average
plasma flow velocity
$\mathbf{u}(t)=\frac{m_{\mathrm{e}}\mathbf{u}_{\mathrm{e}}(t)+m_{\mathrm{i}}\mathbf{u}_{\mathrm{i}}(t)}{m_{\mathrm{e}}+m_{\mathrm{i}}}$
is zero, and the local magnetic field ${\bf B}_{0}$ – averaged over each 30 ms
FPI measurement – is aligned with the $z$-direction. We define a spherical
grid-of-interest in this reference frame, of arbitrary extent and resolution.
For each FPI measurement of the VDF, this grid-of-interest is rotated and
shifted to the instrument reference frame (cf. A) using the measured
background magnetic field ${\bf B}_{0}$ and flow velocity, as well as the
probe motion. The VDF is then interpolated at each node of the transformed
grid, using a tricubic interpolation scheme, documented and tested in B. The
use of a spherical grid allows us to represent an averaged VDF in the
$(v_{\perp},v_{\parallel})$-plane without the need of a binning process, which
is a source of systematic artifacts when working with multidimensional data.
These aspects are illustrated in A. In Figure 1, the left-hand polar plot
shows the result for 10 time-averaged VDFs, with the parallel velocity given
along the vertical axis. A regular spherical grid-of-interest of 200x200x200
nodes was used with a maximum extent of $1.5\cdot 10^{7}$ m/s. To further ease
the reading of the plots, a filled-contour representation is used.

Our goal is to study how the shape of the VDF may be affected by the presence
of a wave. We wish to go further than the reduced description of the VDF given
by its first order moments, namely its number density, its number flux density
($\sim$ bulk velocity) and its momentum flux density tensor ($\sim$
temperature in the thermal equilibrium case). For this purpose, we need a
process that enhances potential patterns which might be “hidden” by the order
zero distribution, or _background_ distribution (not necessarily
Gaussian/Maxwellian). In the example used in Figure 1, the VDF stretches over
more than 6 orders of magnitude in the covered velocity space. The most
obvious, order-0 shape found in the raw, original VDF is a centered, somewhat
isotropic peak. A closer look may reveal obvious departure from the isotropy,
with noticeably straight isocontours for parallel velocities around 0 m/s. We
want a process which highlights these characteristics.

In a numerical context, such a background, or equilibrium distribution $f_{0}$
is usually subtracted from $f(t)$, with the similar goal of enhancing higher
order features (in such a case, the departure from $f_{0}$). It is often
defined as the initial VDF $f(t=0)$, and sometimes as the time averaged
distribution $<f(t)>$. But the resulting _fluctuating_ distribution $\delta
f(t)=f(t)-f_{0}$ might not be a valid concept when dealing with observations,
as the _background_ , order-0 distribution itself is generally varying –
slightly or significantly – during the time interval of interest. In other
words, as the plasma flows and the spacecraft moves, we never probe plasmas
with the same parameters, which may be the case in a controlled simulation
box. Using such a $f_{0}$ with observations results in significant patterns in
velocity space, which should be avoided for studying instantaneous, fine
details of the VDF.

With this motivation, we propose two different treatments of the VDF which do
not rely on any other information than the instantaneous distribution itself.
These two treatments provide two complementary views of the VDF, with
different advantages and drawbacks discussed in the following sections. The
first treatment is a _scaling_ , during which we consider each energy level of
the spherical grid-of-interest separately. For each of these spherical shells,
the minimum value of the interpolated VDF is set to 0 and the maximum value
set to 1. Values are then averaged over the gyro-angle (angle around the
background magnetic field). This scaling is closely related to classical
pitch-angle distributions, in which the VDF is shown for a few selected energy
ranges, with the color-plot dynamics ranging from the lowest to the highest
distribution value of each energy range (see also B). In the proposed scaling,
we virtually display 200 concentric, time-averaged pitch-angle distributions.
In the next section, we will use a _scaled_ pitch-angle distribution, by
selecting only one limited energy range of the scaled VDF and plotting it over
time (Figure 5).

In the second treatment, each VDF values is _normalized_ to a reference value,
which we choose to be the VDF value at the same energy for $v_{\parallel}=0$
(i.e., a pitch-angle of $90^{\circ}$). All these central (equatorial) values
are therefore set to one, appearing in white tones, while higher values appear
as red tones and lower values as blue. A decimal logarithm is applied to the
normalized values when plotted.

These two treatments do not have a physical motivation, they are only
arbitrarily chosen to highlight VDF features.

Figure 1: An example of an original VDF to the left, integrated over 300 ms,
its scaled view in the middle, and its normalized view to the right.

In Figure 1, the effect of the scaling on the original distribution is
remarkable, with two parallel branches of values larger than 0.5 (i.e. red
tones) stretching at constant parallel velocities. Properly speaking, these
two structures are thick disks in the 3-dimensional velocity space. Another
structure is found along the anti-parallel direction, a strahl-like structure
greatly highlighted in comparison with the original VDF. The circular features
of constant speed are unavoidable artifacts of this scaling, in which all
energy levels are treated regardless of the others: the continuity of the VDF
across energy is lost, and some energy shells are scaled differently compared
to the neighbouring shells, resulting in circular visual artifacts. The
normalization we introduced here-above was applied on the original VDF, in
order to conserve this continuity across energy, for a complementary
representation given in the right-most plot of Figure 1. The circular
artifacts vanished from the two branches – or disks – of higher density, which
are now reaching to even higher perpendicular velocities. In this view, the
strahl-like component is perceived as broader than in the scaled VDF. The
normalization puts different weights on details compared to the scaling, and
though being much less sharp, it appears in this case to be more sensitive to
details at high velocities.

We now have a comprehensive and constraining way of examining an instantaneous
VDF, with two different views of it. The 0-to-1 scaling and the normalization
have different properties, already illustrated above and further appreciated
in the following test-particle approach, which makes them a great tool for
comparing numerical and observational results. These two methods are easy to
implement and come with barely any computational cost. Together with the
interpolation approach, they may offer interesting applications for
characterising the multi-dimensional VDF in other heliophysics and planetary
physics contexts.

##  3 Wave analysis and theoretical linear model

Figure 2: (a)-(b) B-field and E-field power density spectrograms. The electron
gyrofrequency is indicated by the solid lines ($f_{ce}$ and $0.4\ f_{ce}$).
(c) Filtered E- and B-field wave forms. (d) Original B-field Power Density
Spectra and its Butterworth filtered copy.

In order to study wave-particle resonances in observational data, we have
isolated one case displaying a clear wave activity within an otherwise “quiet”
magnetosheath environment. Captured on the 8th of March 2019, the 3 minute-
long interval starting at 13:56:10 displays contrasted, strong wave packets
seen in the power density spectrograms of Figure 2 (a-b). The central
frequency of these packets is about 250 Hz, getting slightly higher on the
second half of the observation. It is comprised between 0.3 and 0.5 electron
gyrofrequency $f_{ce}$.

We focus on the central event of constant frequency, selected in Figure 2
(a-b). The fields components are band-pass filtered using a Butterworth
filter. The result is displayed in Figure 2 (d). The reference frame is
aligned with the background magnetic field $\mathbf{B}_{0}$ averaged over the
duration of the selected interval. We find that the parallel component of both
fields is significantly smaller than the perpendicular components, however not
null. In this frequency range, magnetic fluctuations up to 0.5 nT are observed
around an average background magnetic field 25 nT strong (Figure 2 (c)). The
electric field fluctuations are seen sometimes surpassing 1 mV/m.

Figure 3:  One wave packet, indicated in Figure 2 (c) with a grey background,
its analytical model, at position (0,0,0), and its hodogram in the minimum
variance analysis frame. The first data point of the series is indicated by a
cross.

We zoom-in once more to study a single wave packet, highlighted with a grey
background in the time series of Figure 2 (c) and expanded in Figure 3. The
background magnetic field $\mathbf{B}_{0}$ is now calculated over this shorter
period. The fields components exhibit remarkably clean sinusoids within the
envelope defining the packet. The two perpendicular ($x$ and $y$) components
of both fields are almost equal in magnitude and separated in time by a phase
of $\pi/2$, corresponding to an almost perfect right-handed circular
polarisation. The polarisation is also nicely seen in the hodogram of Figure
3, given in the local minimum variance reference frame (the frame in which the
variance along an axis – here the $z$-axis – of the vector time series is the
smallest, 65 times smaller than the two other variances, almost equal to each
other).

Whereas the four perpendicular components are defined by a similar envelope in
the fields $\mathbf{E}$ and $\mathbf{B}$, the parallel (blue) component, also
sinusoidal, follows a different time variation in $\mathbf{E}$ and
$\mathbf{B}$, and has a slightly higher frequency compared to perpendicular
components, in both fields (not shown). Given its right-handed circular
polarisation and its non-zero parallel component, we identify the wave as a
quasi-parallel whistler mode wave. Studying the same mode, lacombe2014apj and
stansby2016apj report its observations in a range from 0.1 to 0.6 gyro-
frequency, a range in which the present case falls perfectly. Because of the
wave right-handed circular polarity and its frequency, electrons are the
species of interest for this study.

We use the following dispersion relation of the whistler mode, derived from
the Appleton-Hartree relation [Hsieh Omura (2017)] in the limit of
$\omega\ll\omega_{pe}$, valid in our case with $\omega<10^{-2}\omega_{pe}$:

$Y=\frac{X^{2}}{X^{2}+1}\cos(\theta)$ (1)

with the normalized spatial frequency $X=k\cdot d_{e}$, the normalized angular
frequency $Y=\omega/\omega_{ce}$, $k$ the amplitude of the wave vector,
$\omega$ the wave angular frequency, $d_{e}$ the electron inertial length,
$\omega_{pe}$ the electron plasma frequency, $\omega_{ce}$ the electron
(angular) gyrofrequency, and $\theta$ the angle between the background
magnetic field and the wave vector.

Because of apparent monochromatic nature of the observed waveforms, we can
determine the wave vector angle $\theta$ with regard to the magnetic field
using a minimum variance analysis over a sliding window as wide as the wave
temporal period. In other words, we consider one pseudo-circle at a time,
described by the wave vector, and find the orientation of the plane containing
this circle. The result of this sliding minimum variance analysis is shown in
the fourth row of Figure 3 for the magnetic field. We find that the wave is
propagating with angles between 0 and 20 degrees from the background magnetic
field. Finally, to determine if this quasi-parallel wave is propagating along
or against the background magnetic field, we calculated the components of the
Poynting vector $\mathbf{E}\times\mathbf{B}$ , which is parallel to the wave
vector (see for instance the comprehensive work of stansby2016apj on single-
spacecraft estimation of whistler mode wave properties). The Poynting vector
$\mathbf{S}$ (Figure 3 third row) is mostly parallel to the background
magnetic field, as evidenced by its largest and positive parallel component:
the wave propagates in a mostly parallel direction. We now have the frequency
and the wave vector direction of the wave, and an estimation of the wave
vector amplitude. It was also verified that the probe velocity in the plasma
frame (in which $\mathbf{u}=0$, see Section 2) is not significant with regard
to the phase speed of the wave, and the observed wave frequency found in the
spectrograms is not Doppler affected.

Despite how clean the observed waves are, they are still limited to one point
in space. If we are to study the possible resonant interactions between the
particles and this precise type of wave, we need its temporal _and_ spatial
description, or model. hsieh2017jgr developed an analytical expression for the
components of a quasi-parallel whistler mode wave, based on the electric field
linear system proposed by stix1962 and obeying the Faraday’s law. With the
$k$-vector lying in the $(x,z)$-plane, the wave fields are given by the
authors as

$\begin{split}\mathbf{B}_{w}&=\mathbf{e}_{x}B_{x}^{w}\cos(\Psi)+\mathbf{e}_{y}B_{y}^{w}\sin(\Psi)-\mathbf{e}_{z}B_{z}^{w}\cos(\Psi)\\\
\mathbf{E}_{w}&=\mathbf{e}_{x}E_{x}^{w}\sin(\Psi)-\mathbf{e}_{y}E_{y}^{w}\cos(\Psi)+\mathbf{e}_{z}E_{z}^{w}\sin(\Psi)\\\
\Psi&=\omega t-k_{x}x-k_{z}z\end{split}$ (2)

Following hsieh2017jgr in the limit $\omega\ll\omega_{pe}$, we get the
following polarizations

$A_{s}=Y+\frac{Y^{2}-1}{\cos(\theta)-Y}\quad;\ \
A_{p}=\frac{\sin(\theta)\cos(\theta)}{\sin^{2}(\theta)-1+\cos(\theta)/Y}$ (3)

Finally, the wave fields components are expressed as

$\begin{gathered}B_{y}^{w}=A_{s}(1-A_{p}\tan(\theta))B_{x}^{w}\quad;\ \
B_{z}^{w}=\tan(\theta)B_{x}^{w},\\\
E_{x}^{w}=A_{s}v_{p\parallel}B_{x}^{w}\quad;\ \
E_{y}^{w}=v_{p\parallel}B_{x}^{w}\quad;\ \
E_{x}^{w}=A_{s}A_{p}v_{p\parallel}B_{x}^{w},\\\
B_{x}^{w}=\frac{B_{w}}{\sqrt{\cos^{2}(\Psi)+A_{s}^{2}(1-A_{p}\tan(\theta))^{2}\sin^{2}(\Psi)+\tan^{2}(\theta)\cos^{2}(\Psi)}},\end{gathered}$
(4)

with the parallel phase speed $v_{p\parallel}=\omega/k_{\parallel}$ .

The observed and modeled wave and plasma parameters for the packet shown in
Figure 3 are gathered in Table 1. Using these parameters, we have estimated
the wave vector amplitude using Equation 1, also given in the same table. We
can now obtain the fully analytical, temporal and spatial model of the wave,
using Equation 4. We note that no information about the electric field is fed
to the analytical model, this field is thus completely constrained by the
model.

The result for a fixed point in space is given in the lower-right panels of
Figure 2 over a few wave periods. The analytical model results in a slightly
higher $E_{w}/B_{w}$ ratio, but provides satisfactory wave forms and Poynting
vector form and amplitude, additionally obeying Maxwell’s equations. We can
now use this model to investigate resonant wave-particle interactions.

| Observed | modeled
---|---|---
$B_{0}$ | 25.4 nT | 25.4 nT
$n_{e}$ | 16.3 cm-3 | 16.3 cm-3
$B_{w}$ | [0.2, 0.7] nT | 0.5 nT
$E_{w}$ | [0.5, 1.4] mV/m | 1.5 mV/m
$\omega\ (\omega/\omega_{ce})$ | 1571 rad/s (0.35) | 1571 rad/s (0.35)
$k$ ($k\cdot d_{e}$) | $\emptyset\ (\emptyset)$ |  $5.9\cdot 10^{-4}$ rad/m (0.77)
$\theta$ | $[0^{\circ},20^{\circ}]$ | $20^{\circ}$
$T_{\parallel}$ | 45 eV | 45 eV
$T_{\perp}$ | 41 eV | 45 eV
$\beta_{e}$ | 0.46 | $\emptyset$

Table 1: Wave parameters for the observation and the model, and additional
plasma parameters.

##  4 Resonant test-particles vs. observed VDF

We will now examine the potential effect of the modeled wave on the electron
dynamics, and see whether or not signatures in the electron VDF can be found.
Before considering simulating the situation with a self-consistent model
(fields and particles feedback on each other according to Maxwell’s
equations), we explore the possibilities offered by a test-particle approach.
The fields and waves are analytically described with constant frequency and
wave-vector through time: there is no feedback from the particles on the
fields. Thus, the presence of other particles, or the shape of the
distribution, does not affect a single particle dynamics, which can be solved
on its own. This allows for a very cheap, flexible first study of the resonant
dynamics in the case of our simplified wave. A considerable advantage is that
by doing so, we isolate the effects _of_ the wave _on_ the particles, and not
the other way around, which is not directly possible in a self-consistent
description.

In the absence of collisions and gravity, and using the analytical wave
discussed in the previous section, we know the force experienced by a particle
at any time and any position. The particles dynamics are solved according to
this force using the classical Boris scheme [Boris (1970)], widely used by the
Particle-In-Cell community because of its straightforward implementation and
its great accuracy for this precise problem – charged particles moving in
electromagnetic fields. We initialise 200 million particles following a
3-dimensional isotropic non-drifting Maxwellian distribution, characterised by
the observed electron (parallel) temperature of 45 eV. These particles are
homogeneously distributed in physical space, in a 2-dimensional periodic box,
extending over one projected wavelength (parallel for one axis, perpendicular
for the other). Results are given for a simulation time of 25 ms,
corresponding to a bit more than 6 wave periods and about 15 electron
gyrations.

Figure 4: Test-particle results (top) compared to the observation (bottom),
for an integration time of 60 ms. The Landau parallel resonant velocity is
given by the horizontal dashed line and the cyclotron one by the solid lines.

The Landau resonance is obtained for particles with a parallel velocity close
to the wave parallel phase speed: seen from these particles’ perspective, the
parallel component of the wave is almost static, and continuously accelerates
them. This gives us a first resonant speed, $v_{Landau}=\omega/k_{\parallel}$.
Because of their angle from the background magnetic field, the observed and
modeled waves have a projected parallel component, which therefore enables
this resonance. Particles with a parallel speed slightly slower than the
parallel phase speed are being caught up by the wave and gain kinetic energy,
whereas faster particles experience the opposite phenomenon. These resonant
particles migrate in phase space, with a motion dependent on their initial
phase space position. The wave we consider here has a dependence along the
perpendicular direction: as a particle gyrates, its perpendicular position
evolves, and so does the magnetic and electric wave components it experiences.
A deeper, 3-dimensional description of these resonant motions, depending on
$v_{\parallel}$ and $v_{\perp}$ is beyond the scope of this article, despite
its great interest. These dynamics quickly result in a mixing of the particles
in phase space around the parallel Landau speed. It is noteworthy that if one
initialises the simulation with a flat velocity distribution (i.e. no density
gradient in velocity space), no signature of this mixing can be found in
velocity space. A density signature is indeed only visible if the resonant
speed corresponds to high VDF gradients, which happens to be the case here.
Very quickly, in just 6 wave periods, the resonant particles form an
under/over-density centered on the Landau resonant speed, as seen in the
scaled and normalized VDF of Figure 4 (the Landau resonant speed is displayed
by the dashed horizontal line).

The cyclotron resonance is only met in the presence of a circularly polarised
wave. Particles with different parallel speeds experience a wave with a
different frequency, a simple Doppler effect dependent on the particles
velocity. For one particular parallel speed, given by
$v_{Cyclo}=(\omega-\omega_{ce})/k_{\parallel}$, with $\omega_{ce}$ the
electron gyrofrequency, the wave is seen with an angular frequency equal to
the electron (angular) gyrofrequency: the particles gyrate synchronously with
the rotation of the circularly polarised $\mathbf{B}_{w}$ and
$\mathbf{E}_{w}$. Note that because $\omega<\omega_{ce}$, this resonant speed
is negative (if $k_{\parallel}>0$): resonant particles are moving against the
wave, which in turn significantly decreases the time during which they may
interact with the wave. As discussed in the Landau case, such particles are
continuously accelerated by the wave, and migrate in phase space, in an even
more complex manner. The mixing of a high density gradient region again
results in an under/over-density organised around the cyclotron speed given by
the solid line in Figure 4, with a sign opposite to the one of the Landau
resonant speed. The over-density is found for lower parallel speeds but higher
perpendicular speeds, corresponding to higher total speed for these particles.

The lower row of Figure 4 gives a comparison to the VDF observed during the
same wave packet. We use only two FPI observations, corresponding to an
integration time of 60 ms. There as well, two over-densities are found just
above the two resonant speeds, at the same location as for the test-particles.
The scaled view of the test-particle VDF exhibits two strong, perfectly
circular artefacts, which were discussed in the first section. The normalized
view, conserving the continuity along the radial dimension, does not display
such structures. An additional feature is also found along the anti-parallel
direction. We will see in the next section that this beam does not correlate
with the presence of the wave. The thermal parallel speed is indicated in the
observed distributions. The speed is the same as the central speed of the
over-densities: the resonant mixing of particles indeed happens where the VDF
velocity gradient $\partial f/\partial v_{\parallel}$ is high, which could
lead to an efficient damping of the wave. We note however that the relevance
of this thermal speed is limited, when considering the large departures of the
observed VDF from a Maxwellian distribution.
The over densities are observed around similar absolute parallel velocities,
despite the absolute value of the resonant parallel velocities being clearly
different: the over-densities tend to be fairly symmetric with regard to the
$(v_{\parallel}=0)$-plane. Note that this fact holds for the plasma and wave
parameters analysed here. If the position of the two over-densities are
matching nicely between the test-particles and the observed VDF, strong
discrepancies in the shape of these signatures exist and are discussed below.

##  5 VDF time evolution

To verify if this double branch signature indeed correlates with the presence
of the wave over longer time scales, we show In Figure 5 (c-d) two additional
electron VDFs integrated over a longer time of ten FPI measurements,
corresponding to an integration time of 300 ms. In both cases, a strahl-like
(beam) component is found along the anti-parallel direction, with a constant
perpendicular width. In the absence of wave activity (last VDFs), we find a
clearly anisotropic distribution, with a strong equatorial signature in the
scaled view.
When analysed during the maximum power of the wave activity, the VDF displays
the double-branch signature. It was checked that during the entire time
interval, no significant (i.e. higher than the measurement noise) agyrotropy
are to be found. Ions (not shown here) present a similarly anisotropic
distribution, virtually static over the time interval.
Figure 5 (b) gives a view on the temporal evolution of the scaled VDF and its
double-branch feature. For this representation, each 30 ms scaled VDF was
averaged over a limited speed range between 5.8 and 6.3 $10^{6}$ m/s
(indicated on the scaled view in panel (c)), and the result was plotted in the
time $t$ and pitch-angle $\Theta$ dimensions in panel (b). Note that scaled
pitch-angle distributions and classical pitch-angle distributions
qualitatively converge as the energy range decreases. In this energy range,
scaled values above 0.5 (red tones) form a striking feature centered on 90
degrees. When the waves are observed in the magnetic field spectrogram (panel
(a)), the feature splits into two branches, corresponding to the double-branch
pattern observed in velocity space. We note that such a pitch-angle
distribution looks very similar to electron distributions in mirror modes,
explored by breuillard2018jgr, where the envelope of the pitch-angle
distribution is shown to correlate with a critical pitch-angle, under which
electrons get trapped in one mirror structure. We note, however, that in our
case, the signature is ordered by (constant) parallel velocities, and not
constant pitch-angles when all the energy range is considered.

Figure 5: (a) B-field power density spectrogram. (b) scaled pitch-angle
distribution for a speed range indicated in (c). (c)-(d) time-integrated VDFs
and their scaled view, with the integration time indicated on the time series.

##  6 Particle energisation

We can follow the energisation of the test-particles either by summing up the
total kinetic energy of all particles, or simply by taking the center of mass
of their distribution (order 1 moment) and expressing it as a current density,
with one parallel and one perpendicular component. The total kinetic energy
and the two components of the current density are given in Figure 6. We find
that with the physical parameters of our problem, the maximum energisation
happens very fast, in about 2 wave periods, or 5 gyroperiods. Therefore, the
wave packets are sufficiently long to exchange energy with the particles.
After about 5 wave periods, the overall energy gained by the test-particles,
as well as the parallel current density, stabilise to a non-zero value. The
test-particle approximation does not hold anymore, as this current would
necessarily alter the electric and magnetic fields, which would act to
decrease it back to zero. For this reason, it is likely that this estimated
current for the present resonant wave-particle interaction is an upper bound
for the self-consistent interaction. Barely any perpendicular current is
found, indicating that the VDF remains gyrotropic at all times. The fact that
the total kinetic energy of all particles and their parallel current have
almost the same time evolution illustrates that these resonant interactions
mostly result in an increase of the bulk velocity and only some of the energy
is transformed into “temperature”, or change of the pressure tensor.

A crucial point is indicated by this time evolution and verified in the time
evolution of the simulated VDF (not shown) is that the resonant signatures are
not periodic: they remain at the same position (in velocity space) with the
same over/under densities through time. This is a necessary condition for it
to be observed by an instrument with an integration time longer than the wave
period.

Figure 6: Test-particle total kinetic energy (relative to its value at t=0)
and current density, parallel and perpendicular. Time is given in unit of wave
period $T_{wave}$.

In Figure 7, the observed electron current density (parallel and
perpendicular) are displayed, together with the cropped spectrogram of the
magnetic field. The parallel current is varying around 0, and no correlation
with the wave activity can be found. Despite the strength and duration of the
wave packets, the _background_ current density turns out to be significantly
larger than the estimated (upper bound) current rising from the resonant
particle signatures. Alternative moments were also calculated ignoring the
core of the distribution (using different speed threshold), showing barely any
changes in the electron current. In turn, the dissipative term
$\mathbf{E}\cdot\mathbf{J}_{e}$, using an electric field averaged over the
electron integration time, shows no correlation with the wave activity.

Figure 7: (a) Magnetic field spectrogram, (b) observed electron current and
(c) dissipative term.

##  7 Discussion & Conclusions

We have first shown that VDF features originally hidden under high gradients
of the background VDF can be highlighted using some scaling and normalization,
without the use of a reference VDF $f_{0}$. We then isolated a strong, fairly
narrow-band, quasi-parallel whistler mode wave, which could be analytically
mimicked. We applied this analytical wave to a collection of test-particles to
get a first sense of its potential effects on the electrons. We could map
where and how particles resonate, and have found that the initial Maxwellian
distribution is reshaped by Landau and cyclotron resonances. Test-particles
and observed VDFs display two branches of higher density, around constant
parallel velocities, in the scaled and normalized views of the VDF. The
observed signatures were found to correlate nicely with the wave activity.
Finally, this wave-particle interaction could not be detected in the observed
current or the observed $\mathbf{E}.\mathbf{J}_{e}$ product, as both
$\mathbf{E}$ and $\mathbf{J}_{e}$ (observed) present fluctuations of larger
amplitude than the current and electric field used by or resulting from the
model. Strong discrepancies between the simulation and the observations remain
to be discussed.

Firstly, the strongest discrepancy between the observed VDF and the simulated
one is the strong symmetry of the observed signatures around the
$(v_{\parallel}=0)$-plane, further illustrated by the time evolution of the
scaled pitch-angle distribution in Figure 5. As such, it is difficult to prove
that the Landau resonance, providing an already faint signature in the perfect
test-particle set-up, may cause a resonant branch as strong as the
observation, and so symmetric to the cyclotron branch. It actually _appears_
as if a mirrored cyclotron branch shows up systematically with the wave
activity, despite our wave analysis only bringing out one single mode,
propagating in one well defined direction. Such symmetric signatures have been
displayed in the works of min2014jgr (in the magnetosphere) and chen2019nc (in
a nominal, or fully developed turbulence case), in which the authors did not
isolate one single mode, and interpret the symmetric signatures as caused by
waves propagating in both directions, parallel and anti-parallel. This
symmetry remains the foremost open question of our study.

Secondly, the test-particle model cannot account for the transformation from
the already strongly anisotropic VDF out of the wave activity (shown in panel
(d) of Figure 5) to the 2-branch signature. The disturbance of the VDF caused
by resonances in the test-particle case is a local phenomenon in velocity
space, it certainly cannot reshape macroscopically the distribution, make a
large amount of the electron population migrate over large velocities. Were we
able to easily initialise the particles according to the observed VDF outside
the wave activity, the strong equatorial structure would remain there, with
additional resonant signature expressed on the sides. While the test-particle
simulation enlighten us on the micro-physics of the interaction during a very
brief snapshot of the observations (when an intense wave activity is
observed), the macroscopic configuration of the fields (magnetic gradients,
mirrors, etc) should be considered to fully understand the time evolution of
the electron VDF.

The limitations of the test-particle simulation are numerous, if one is to
thoroughly compare its results and the observations. Instead, we suggest that
this approach only points at the very first step of the full self-consistent
wave-particle interaction. It shows us where and how in velocity space
resonances should occur for a given a monochromatic wave, and gives a first
hint on their phase density signatures. Most importantly, it also gives us a
good sense of how fast these resonances can produce signatures in the VDF, and
their efficiency. But it obviously cannot go further in the physics of the
interaction, both an advantage and a drawback.
This numerical approach is limited to the case of a purely monochromatic wave
with a constant amplitude and a constant normal angle, whereas the observed
wave packets exhibit some spectral breadth, a parallel component at a slightly
higher central frequency, and additionally a propagation direction constantly
evolving. Because of the periodic boundaries necessary for the simple test-
particle approach, such a wave cannot be easily modeled. We refer to the work
of tao2012grl, in which the authors specifically highlighted the effect of the
amplitude modulation of chorus emissions in the magnetosphere, on particle
acceleration. They, however, do not reconstruct distribution functions, and
can therefore do without the periodicity constraint.

The feedback of the particles on the wave is a different problem entirely, in
terms of numerics at least. Thus the comparison should not be over-
interpreted. We have shown that the observed wave is expected to have clear
signatures on the electron distribution function, signatures matching well in
position and somewhat in shape the observed signatures.

The visualisations (scaling and normalization) developed for this study are a
key element for a better, deeper characterisation of the particles
distributions. It is in the numerical aspect of our work that these views
reveal best the effect on the distribution, getting us rid of the need for an
additional information (e.g. distributions at other times). They show great
flexibility and versatility, which may be useful for other applications on
observations and simulation data, and mostly their comparisons.

Such an exploration of wave-particle resonances in 3 dimensions, and its
direct comparison to observations, is – to the best of our knowledge – novel.
It familiarises us with resonant dynamics and signatures, at a low cost and
without the intrinsic complexity a self-consistent model adds to physical
interpretations. This approach may present real promises for a more systematic
recognition of resonant signatures in a more complex data, such as those of
fully developed turbulence in the solar wind or the magnetosheath

###### Acknowledgements.

E. Behar is funded through DIM-ACAV post-doctoral fellowship and from the
European Unions Horizon 2020 research and innovation program under grant
agreement No 776262 (AIDA, www.aida-space.eu). The authors also wish to thank
Olivier Le Contel and Gérard Belmont for valuable discussions that have
broadened the context of this study.

##  Appendix A Binning versus interpolating

Figure 8: Illustration of a binning and an interpolation.

In various publication, when handling multidimensional Velocity Distribution
Functions (VDF), authors choose the somewhat more intuitive of rotating the
data from the instrument frame to the frame of interest, to then _bin_ the
data within a defined grid, as illustrated in Figure 8 in the first row.
During this process, one measurement point ends up in one single cell of the
grid-of-interest, and one cell of the grid-of-interest may receive zero one,
or many measurements. Empty cells will appear on the representation, producing
visual artifacts, and it is necessary to average the binned data over at least
one dimension in order to _fill_ as much as possible the grid-of-interest. We
give such an example in a cylindrical representation of the VDF, right-most
plot of the first row of Figure 8, which shows about thirty electron
distributions binned in a 2-dimensional grid defined, in the classical
$(v_{\parallel},v_{\perp})$-plane. The binned data show strong artifact along
the energy dimension, as the instrument energy levels are log-distributed: the
higher the energy, or speed, the greater the energy steps. Therefore one
regular Cartesian grid cannot be suitable for the entire energy range of the
instrument: its resolution will be too coarse at low energies – where
weighting of the binned data has to be taken into account properly – and too
fine for high energies, leaving empty most of the grid-of-interest. The
binning approach can only be used over a restricted range of energies, and is
anyway not suitable for 3-dimensional analyses.

The interpolation approach is illustrated in the second row of Figure 8.
First, we define a 3-dimensional grid-of-interest – an array/set of
coordinates – within the reference frame of our choice, with arbitrary extent
and resolution. We apply the opposite rotation, shift, and scaling to a copy
of the grid-of-interest, from the reference frame to the instrument frame, as
illustrated in Figure 8. The values of the data are then interpolated at each
node of the transformed grid (see next section): the data are continuously
evaluated over the entire grid, leaving no room for artifacts, and avoiding
additional weighing operation. All the results in the article use a spherical
grid-of-interest, allowing at no additional cost the scaling and normalization
of the VDF, and their straightforward representation in the
$(v_{\parallel},v_{\perp})$-plane.

But interpolation with an order higher than 2 presents complications, and this
approach is not universal and cannot be applied blindly.

##  Appendix B Interpolating velocity space distributions

We have tested three different interpolation schemes, namely nearest-neighbour
(order 0, only one point of the measured VDF is used), trilinear (order 1, 8
data points are used), and tricubic (order 2, 64 data points are used). The
two first schemes are taken from the main scientific python library, SciPy,
while the third was implemented by [Lekien Marsden (2005)], with one wrapper
made available by the authors for a usage with Python.

Figure 9: Interpolation.

We now have a 3-dimensional array of interpolated VDF values, with two
different sets of coordinates, one in the instrument frame (used for the
interpolation), one in the reference frame of the study (the originally
defined Grid-of-Interest). This array can be either analysed and visualised on
its own, or added up to other arrays in order to average the data over a
longer duration, all within the same frame. For instance, Figure 9 presents
results using only one single electron VDF, which is the most challenging test
for the methods, because of low detection rates for higher energies.

This method has one invaluable interest: any resolution can be used without
any of the risks inherent to a binning process, illustrated in Figure 8.
Another great advantage when using a Cartesian grid-of-interest is that
velocity derivatives of the data can be easily and straightforwardly computed.
The advantage of using a spherical or cylindrical grid-of-interest was already
illustrated. The method has one drawback, namely the computational cost. The
interpolation in itself is more computationally demanding than a simple
binning, and for most purposes we will have many more elements in the grid-of-
interest than in the instrument grid, resulting in as many more calculations
to compute.

In Figure 9, a single electron distribution is used to illustrate the
differences between the three interpolation schemes, using three different
visualisations. In the first row, we give the profile (1-dimensional) of the
interpolated values along the parallel direction. For this representation and
in order to increase slightly the statistics, all values within an angular
distance from the parallel axis ($+\mathbf{B}$) are selected and averaged,
using a conic selection $\pi/6$-wide. For this purpose, a spherical grid-of-
interest is a better choice, allowing one to simply average the interpolated
data over one angular dimension, avoiding a binning process.

The energy/speed levels of the instrument are also indicated as vertical
lines, regularly spaced in this logarithmic representation. For this analysis,
we only rotated the frame, without a shift, so the energy levels of the
instruments remain centered on the origin of the Grid-of-Interest, making the
results more readable.

Another representation is proposed on the second row of Figure 9, in which we
selected a specific speed range, averaged the data over this range, resulting
in the given angular maps (2-dimensional). Here as well, using a spherical
Grid-of-Interest makes this selection straightforward, selecting only one
radial range in the array, again avoiding a binning process. The parallel axis
is vertical and intersects these maps at their poles, and the equator
correspond to a pitch angle of 90 degrees.

In the parallel 1-dimensional profiles previously described, the nearest-
neighbour interpolation results in steps centered around the instrument energy
levels, as expected from such a scheme. This curve is reported in the two
other VDF profiles for the trilinear and tricubic cases, in order to verify
that all three interpolations indeed meet at the instrument speed levels, an
important convergence test. These steps in the nearest-neighbour case directly
correspond to the “pixels” found in the left-most angular map. This is the way
particle data are often displayed, implying that within an instrument “pixel”,
the observed flux is constant. This assumption is also widely used when
integrating the plasma moments, which simply corresponds to a Riemann sum of
the area under a function. The instrument poles are misaligned with the
magnetic field direction, and can be seen close to the equator, 180 degrees
apart, with triangular pixels meeting in one point.

The trilinear interpolation provides a smoother, more continuous profile.
Thinking in one dimension, between two measurement points, interpolated values
will follow a linear relationship. In this logarithmic representation, these
linear segments show up as arc segments in-between each instrument
energy/speed level. This is obviously an unwanted visual artifact, generated
by the choice of the representation. Just as the 1-dimensional profile, the
angular map shows a smoother result than in the nearest-neighbour case. In the
same way that the nearest-neighbour can be linked to a Riemann sum, this
linear interpolation converge to a trapezoidal rule in terms of VDF integral.

Finally, the tricubic interpolation provides the smoothest and most continuous
curve for speeds up to 7000 km/s, for which the VDF is relatively high. At
higher energies, strong artifacts are found, with oscillations between the
last instrument speed levels. Two phenomena can account for this behaviour.
The first could be the Runge’s phenomenon, namely oscillations at the edges of
the interpolation interval when using polynomials of high degree (higher than
two). We note however that we only use a third degree polynomial
interpolation, which should limit this phenomenon, and these oscillations are
only visually found for low fluxes, high speeds. The second phenomenon occurs
when the VDF contains zero values. But most of all, this interpolation scheme,
as derived and implemented by lekien2005nme, can result in negative VDF
values. This nonphysical result should be monitored, so these negative values
remain small and insignificant for the analysis we want to perform.

_Remark_ : these interpolations and their unavoidable artifacts and drawbacks
are not well suited for all purposes. For instance, the tricubic interpolation
is valid only for strong signals, high VDF values, and should anyway be used
with great caution, so these artifacts are not interpreted as physical
features.

As a first test, we have already verified that the three schemes converge at
the instrument speed levels. A second important test is to make sure that the
three interpolations do not show artifacts at the instrument poles. Indeed,
the trilinear and tricubic schemes we use assume that the data are defined
over a grid with cuboid elements: evenly spaced along each dimension, with
possibly different spacings along each dimension. The poles of the instrument
spherical grid should therefore present errors, as the array elements there
strongly depart from cuboids. To test the overall error from each scheme, we
have defined artificial, ideal instrument measurements, namely a
drifting111The drift is important for the significance of the test, as it
misaligns the center of the artificial distribution, and the center of the
instrument grid, testing the errors over all three dimensions. Maxwellian
distribution. This way, we also know the real value of the distribution at the
nodes of the rotated, shifted, scaled grid-of-interest, and can compare these
analytical values to the interpolated ones. It allows us to creates error
maps, presented in Figure 10. The first row shows one chosen Cartesian cut
through the error distribution, while the second row gives one angular cut, or
angular map. The nearest-neighbour interpolation results in the largest
errors, just as expected from the steps seen in the 1-dimensional profiles of
Figure 9: the interpolation is alternately much smaller and much larger than
the real analytical value of the distribution. These errors can largely
surpass the 50 percent level, because of the steepness of the VDF. Since the
Maxwellian distribution is convex for high energies/speed, the error given by
the linear interpolation is positive-only for high speed values, as seen in
the first row, middle panel. At these higher speeds, the error can reach up to
50 percent, whereas at lower speed values – of greater interest for us – the
error becomes arbitrarily low. The tricubic interpolation shows the smallest
error in this plane with mostly positive values. We note a slightly stronger
error-ring at the lowest speeds covered by the instrument, which we link to
the Runge’s phenomenon. At the high VDF values usually observed at these
energies, this error may become significant for some purposes, though it has
not been found to be the case for our analyses.

The instrument poles are not intersected by this plane, but we expect to find
the greater error there. They however show up in any angular map, as seen in
the second row of Figure 10. The poles still have the same position as
previously, and can be easily spotted in each interpolation error map. The
worst qualitative result is obtained for the tricubic interpolation, with
relatively strong errors localised at the instrument poles, though the
absolute error remains low (a few percents) and indeed extremely localised. We
conclude that for our purpose, the three schemes are sound, resulting in
acceptable errors where the observed VDF values are high, i.e. for speeds
lower than $10^{4}$ m/s.

Figure 10: Error interpolation.

As a third test, we have verified that the electron density, resulting from
integrating the interpolated values using a simple Riemann integral,
calculated for the three interpolations, matches the density provided by the
instrument team. Their densities are also the plasma moment of order 0, but
directly integrated in the instrument spherical coordinate system. The result
is shown in the first row of Figure 11, and one can find that the four curves
are barely distinguishable.

Figure 11: Moments tests.

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