supplemental_material.pdf

# Identification of coupled Landau and anomalous resonances in space plasmas

Jing-Huan Li  Xu-Zhi Zhou<EMAIL_ADDRESS>Zhi-Yang Liu  Shan Wang  School
of Earth and Space Sciences, Peking University, Beijing 100871, China  Anton
V. Artemyev  Institute of Geophysics and Planetary Physics, University of
California, Los Angeles, California 90095, USA  Yoshiharu Omura  Research
Institute for Sustainable Humanosphere, Kyoto University, Kyoto 611-0011,
Japan  Xiao-Jia Zhang  Department of Physics, University of Texas at Dallas,
Richardson, TX 75080, USA  Li Li  Chao Yue  Qiu-Gang Zong  School of Earth and
Space Sciences, Peking University, Beijing 100871, China  Craig Pollock
Denali Scientific, Fairbanks, AK 99709, USA  Guan Le  NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA  James L. Burch  Southwest Research
Institute, San Antonio, TX 78238, USA

###### Abstract

Wave-particle resonance, a ubiquitous process in the plasma universe, occurs
when resonant particles observe a constant wave phase to enable sustained
energy transfer. Here, we present spacecraft observations of simultaneous
Landau and anomalous resonances between oblique whistler waves and the same
group of protons, which are evidenced, respectively, by phase-space rings in
parallel-velocity spectra and phase-bunched distributions in gyro-phase
spectra. Our results indicate the coupling between Landau and anomalous
resonances via the overlapping of the resonance islands.

Suggested keywords

††preprint: APS/123-QED

##  I 1\. Introduction

In collisionless space plasma environments, the dynamics of charged particles
are largely governed by their interactions with plasma waves. The wave-
particle energy transfer becomes especially efficient when they resonate,
namely, the particle observes a constant wave phase. The resonance
condition[1] in magnetized plasma is usually given by

$\displaystyle\omega-k_{\parallel}v_{\parallel}=n\Omega$ (1)

where $\omega$ is the wave frequency, $k_{\parallel}$ is the parallel
wavenumber, $v_{\parallel}$ is the particle’s parallel velocity, $\Omega$ is
the particle’s gyrofrequency, and $n$ is an integer. A fundamental resonant
process, the Landau resonance, applies when $n=0$ so that the particle moves
at the same speed as the wave propagation along the background magnetic field
to observe a constant phase $\phi_{E\parallel}$ of the wave parallel electric
field $E_{\parallel}$. Another important resonance is the first-order
cyclotron resonance, which occurs at $n=1$ as the Doppler-shifted wave
frequency matches the particle’s gyrofrequency. Therefore, the phase
difference $\zeta$ between the particle’s perpendicular velocity and the wave
magnetic field remains constant[2]. Accordingly, a resonant velocity[2] is
defined as

$\displaystyle V_{r}=(\omega-\Omega)/k_{\parallel}$ (2)

to represent the parallel velocity of the particle in cyclotron resonance with
the waves.

As a resonant particle gains or loses energy from the waves, its parallel
velocity $v_{\parallel}$ also changes to deviate from the resonant velocity,
which leads to the variation of wave-particle phase difference
($\phi_{E\parallel}$ for Landau resonance or $\zeta$ for cyclotron resonance).
For some resonant particles, however, the $\phi_{E\parallel}$ or $\zeta$
variations are periodic to form closed particle trajectories in the
$v_{\parallel}-\phi_{E\parallel}$ or $v_{\parallel}-\zeta$ phase space. This
process is called nonlinear trapping, and the phase-space region occupied by
the trapped particles is named resonance island[3, 2, 4]. These nonlinear
wave-particle interactions play an important role in the dynamics of space
environments. In the radiation belts, they are responsible for the frequency
chirping of chorus waves[2, 4] and the acceleration/precipitation of
relativistic electrons[5, 6]. In the foreshock solar wind, they produce
gyrophase-bunched ions[7, 8, 9] and accelerate/thermalize electron beams[10,
11].

The cyclotron resonance can be further modified into anomalous resonance if
the particle’s gyromotion around the background magnetic field is
significantly perturbed by the wave field, which changes the particle’s
angular velocity and thus revises the resonance condition (1)[12, 13, 14, 15,
16]. Moreover, the original resonance island can be dissociated into two
separate islands when anomalous resonance occurs[16]. Interestingly, the
Landau and anomalous/cyclotron resonances may occur simultaneously for
oblique-propagating waves[17], although it remains unclear whether or how they
are coupled.

Spacecraft identification of wave-particle resonance often relies on
estimations of wave-particle energy transfer [18, 19, 20] and/or averaged
distributions of particles indicative of their diffusion processes [21, 22,
17, 23]. Recently, the high-resolution particle observations from
Magnetospheric Multiscale (MMS) spacecraft provide more detailed insights into
particle dynamics within a single wave period. For instance, the observations
of phase-bunched stripes in gyro-phase spectra[24, 25] and V-shaped structures
in pitch-angle spectra[26] shed new light on nonlinear trapping of resonant
particles in the wave field.

In this paper, we use MMS observations to study the interactions between
large-amplitude oblique whistler waves in the terrestrial foreshock and solar
wind protons. In the parallel direction, a series of proton phase-space rings
are observed with a one-to-one correspondence to $E_{\parallel}$ variations
indicative of nonlinear Landau resonance. In the perpendicular plane, the
periodic appearance of phase-bunched ion structures supports the occurrence of
anomalous resonance. This is a direct observational identification of
simultaneous Landau and anomalous resonances, and their coupling demonstrates
the important roles of oblique waves in governing the complicated particle
dynamics in space.

##  II 2\. Observations

###  II.1 2.1 Overview

At $\sim$ 1630 UT on February 5, 2021, the four-spacecraft MMS constellation
was located near the terrestrial bow shock at [13.7, 6.1, -6.1 ] $R_{E}$
(Earth radius) in Geocentric Solar Eclipse (GSE) coordinates (with $\bm{x}$
axis towards the Sun and $\bm{z}$ axis perpendicular to the ecliptic plane).
The onboard instruments offer high-resolution electromagnetic and particle
measurements to analyze the wave-particle interactions [27, 28, 29]. In this
event, the protons contribute over 99% to the plasma density, and we
hereinafter neglect the contribution of heavier ions.

Figure 1 provides a 1.5-hour overview of MMS3 observations. Similar features
are also observed by other MMS satellites (not shown) due to their minor
separations. The energy spectrum of ion energy fluxes in Fig. 1a indicates the
back-and-forth crossings of the bow shock, characterized by the shifts between
narrower and wider energy distributions that represent the foreshock solar
wind and the thermalized magnetosheath ions, respectively. These features also
agree with the higher and lower plasma bulk speeds in the foreshock and
magnetosheath, respectively (see Fig. 1b). The magnetic field $B_{y}$ in Fig.
1c indicates the occurrence of ultralow-frequency (ULF) waves at the period of
30-50s, which have been commonly reported in these regions[30, 31, 32].

Higher-frequency waves are also present in this event. Fig. 1d shows the
magnetic measurements from a 3-min segment, which exhibit oscillations with a
decreasing wave period from 10s to 4s in the shadowed interval. The waves are
also characterized by its large amplitude $B_{1}$, around 1.5 times the
background field $B_{0}$. The shadowed interval is then chosen for in-depth
analysis of wave properties and wave-ion interactions. Similar findings can be
also drawn from observations near the red- and green lines in Fig. 1b-c, as
detailed in the Supplementary Material [33].

###  II.2 2.2 Wave property

Figure 1:  MMS3 observations of oblique whistler waves. The top panels provide
a 1.5-h overview, including (a) ion energy spectrum, (b) bulk velocity, and
(c) magnetic field $B_{y}$. (d) A 3-min overview of magnetic field variations.
The zoomed-in view of the shadowed interval is shown in the bottom panels,
including (e) $\bm{B}_{1\perp}$, (f) $E_{\parallel}$, overplotted by the
integrated potential. (g) $v_{\parallel}-t$ spectra, (h-i) gyro-phase spectra.
The black lines represent the gyro-phase of -$\bm{B}_{1}$.

We first determine, based on the four-spacecraft measurements of the magnetic
field, the wave vectors at each frequency [34]. The phase difference analysis
[34], with details given in the Supplementary Material[33], provides the
measured wave dispersion relation in the reference frame moving with the
spacecraft, which can be further transformed into the plasma rest frame. This
result reveals that the observed wave phase velocity $\bm{v}_{w}$ varies only
slightly within the wave frequency range of interest, which is approximately
$[-84.8\pm 4.1,32.0\pm 5.0,4.1\pm 4.0]$ km/s. The estimated wave normal angle,
approximately $150^{\circ}$ between $\bm{v}_{w}$ and the background field
$\bm{B}_{0}$, demonstrates an oblique wave propagation. The corresponding
parallel wavenumbers $k_{\parallel}$ range from $\sim-0.6/d_{p}$ to
$\sim-1.5/d_{p}$, where $d_{p}\sim 100$km is the proton inertial length. These
estimations are further validated by the application of electromagnetic
singular value decomposition analysis [35], which yields similar results (see
Supplementary Materials [33] for details).

We further establish a field-aligned coordinate (FAC) with the parallel
direction $\bm{e}_{\parallel}$ defined along $\bm{B}_{0}$, which is derived
from a 15s running average of the field measurements. In the perpendicular
plane, the $\bm{e}_{\perp 1}$ direction is along the cross product of
$\bm{B}_{0}$ and $\bm{v}_{w}$, and $\bm{e}_{\perp 2}$ completes the triad.
Based on this coordinate, the particle gyro-phase angle is defined by the
angle between its perpendicular velocity $\bm{v}_{\perp}$ and the
$\bm{e}_{\perp 1}$ axis. The zoomed-in observations of the waves and the
associated proton distributions, organized in the FAC coordinate, are shown in
the bottom panels of Fig. 1. The perpendicular wave field $\bm{B}_{1\perp}$,
filtered from 0.1 to 0.3 Hz (close to proton gyrofrequency $f_{cp}\sim
0.3$Hz), is displayed in Fig. 1e. Obviously, its $\perp_{2}$-component leads
its $\perp_{1}$-component in phase by $\pi$/2, which indicates a left-handed
polarization. In the plasma rest frame, however, the waves propagate sunward
at the speed of $[121.5\pm 5.4,-45.9\pm 6.9,-5.9\pm 5.9]$ km/s, which
indicates a polarization reversal into right-handed waves due to the Doppler
effect (with a frequency around -1.4 rad/s) [26, 32]. The waves are also
associated with $E_{\parallel}$ oscillations (black line in Fig. 1f), and
therefore, are identified as oblique whistler waves frequently reported in the
foreshock [32, 36].

###  II.3 2.3 Wave-proton interactions

To study the wave-proton interactions, we separate the time interval of
interest into five wave periods (see the colored shadows in Fig. 1f) based on
the electric-field phase angle $\phi_{E\parallel}$, which increases from 0° to
360° during each period. The dashed lines mark $\phi_{E\parallel}$=180°, where
the parallel electric potential (blue line in Fig. 1f, integration of the
observed $E_{\parallel}$) minimizes.

Fig. 1g shows the ion phase-space density (PSD) distributions as functions of
$v_{\parallel}$ and time, which display a series of rings with high PSDs
surrounding their lower-PSD centers. All the ring centers are located near the
potential minimum with $\phi_{E\parallel}$ $\sim$ 180° (the dashed lines), and
their parallel velocities ($v_{\parallel}$ $\sim$ -100 km/s) are close to the
parallel wave speed $v_{w\parallel}$ = -105.0 $\pm$ 4.3 km/s. These five
rings, as will be shown in the simulation section, represent the phase-space
trajectories of the ions trapped within the wave-carried potential wells (the
regions surrounding the potential minima), in which the ion parallel
velocities vary between -300 and 100 km/s. The velocity variations, $\pm 200$
km/s in the wave rest frame, are consistent with a 200-V potential well (shown
in Fig. 1f) and the width of the Landau resonance island[1]. In other words,
they are signatures of nonlinear Landau trapping for ions resonating with the
whistler waves.

Interestingly, the same ion population is also modulated in the perpendicular
plane. Fig. 1h-i shows the gyro-phase spectra of the 256- and 199-eV ions at
the pitch-angle range of 130°±20° (corresponding to a $v_{\parallel}$ range
from -200km/s to -100 km/s), which are characterized by periodic occurrences
of inclined stripes with enhanced PSDs. Also shown in Fig. 1h-i (as black
lines) are the gyro-phase of -$\bm{B}_{1}$, which are mostly aligned with the
inclined stripes to indicate the phase-bunched ion behavior at
$\zeta$$\sim$180°. These features are usually treated as diagnostic signatures
of cyclotron resonance[24, 25]. However, based on the wave properties given
above, the cyclotron resonance velocity $V_{r}$ is expected to be $\sim$ 108.7
$\pm$ 58.8 km/s, which differs dramatically from the parallel velocity of the
phase-bunched ions.

Figure 2: Virtual spacecraft observations. Panels a-e are in the same format
as Fig. 1e-1i.

The deviation from the classical cyclotron-resonance theory could be explained
by the occurrence of anomalous resonance (also named anomalous trapping[15],
distinctly different from anomalous cyclotron resonance[37]), which takes
place when the wave-associated forces, including the
$q\bm{v}_{\parallel}\times\bm{B}_{1}$ and $q\bm{E}_{1}$ forces in the
perpendicular plane, are comparable to the background
$q\bm{v}_{\perp}\times\bm{B}_{0}$ force. This is made possible by the
comparable $B_{0}$ and $B_{1}$ magnitudes. The additional, wave-associated
forces modify the conventional gyrofrequency $\Omega$, and consequently,
provide a new term in the classical condition (1)[15, 16]. The resulting
resonance speed (for ions at $\zeta$=180°) [16] equals

$\displaystyle
V_{r}^{{}^{\prime}}=V_{r}-\frac{V_{r}-v_{w}}{k_{\parallel}v_{\perp}/\Omega_{1}+1}$
(3)

where $V_{r}$ is obtained from equation (2), and $\Omega_{1}=B_{1}q/m$ is the
nominal gyrofrequency associated with $B_{1}$. Based on equation (3),
$V_{r}^{{}^{\prime}}$ is estimated at approximately $-200$ km/s, corresponding
to the pitch angle of 130°±20° for 100-500eV ions. Therefore, we speculate
that the phase-bunched ion distributions in Fig. 1h-i are caused by anomalous,
rather than classical cyclotron, resonance. The observations suggest that
oblique whistler waves could interact with the ions via Landau and anomalous
resonances simultaneously, which will be examined through test-particle
simulations.

##  III 3\. Simulation

We next follow the test-particle approach of ref.[16] to study the interaction
between oblique whistler waves and solar wind ions. In the simulation, the ion
velocity distributions at any time and location are determined by tracing the
ions backward in time, since Liouville’s theorem requires PSD conservation
along particle trajectories (see Supplementary Material for detailed
simulation setup[33]). The 3D distributions are integrated to obtain the
$v_{\parallel}$-t and/or gyro-phase spectrograms in the same format as
observations in Fig. 1g-i.

Figure 3: Illustration of phase-space ring formation. (a) $E_{\parallel}$. (b)
$v_{\parallel}$ (c) Initial $v_{\parallel}$ distribution, with black color
denoting high PSDs. (d) Phase-mixing process. (e-h) $\phi_{E\parallel}$ and
$v_{\parallel}$ variations for the green and red ions, respectively.

Fig. 2 shows virtual spacecraft observations of the simulated field and ion
distributions since $t=t_{0}$+45s. Fig. 2a-b display the perpendicular wave
magnetic field $\bm{B}_{1\perp}$ and parallel electric field $E_{\parallel}$,
respectively. Fig. 2c shows the simulated $v_{\parallel}$-t spectrogram of the
ion PSDs, which resembles the periodic appearance of phase-space rings with a
one-to-one correspondence to $E_{\parallel}$ variations in Fig. 2b. The
similarities between simulations and observations (compare Fig. 2c to Fig. 1g)
enable us to understand the processes underlying the phase-space rings via
analysis of the ion trajectories.

A schematic framework of our analysis is shown in Fig. 3. Here,
$E_{\parallel}$ profile is given in Fig. 3a, with its phase angle
$\phi_{E\parallel}$ increasing from 0° to 360° during each wave period. The
corresponding electric potential minimizes at $\phi_{E\parallel}$ = 180°,
where the resonant ions are trapped nonlinearly (Landau trapping, see Fig. 3b
for illustrations of closed trajectories in the
$\phi_{E\parallel}$-$v_{\parallel}$ phase space). The green ion in Fig. 3b is
phase-locked at the center of the potential well ($\phi_{E\parallel}$ = 180°),
which represents the deeply-trapped ions with negligible $\phi_{E\parallel}$
and $v_{\parallel}$ variations. The wave-carried potential well can also trap
the red ion in Fig. 3b despite larger variations in $\phi_{E\parallel}$ and
$v_{\parallel}$. Since its trajectory is close to the separatrix of the
resonance island, the ion belongs to the shallowly-trapped population.

The initial PSDs for deeply- and shallowly-trapped ions are distinctly
different, however. The initial ion distributions in [33] indicate PSD peaks
at $v_{\parallel}\sim-370$ km/s (the background solar wind), which is much
lower than $v_{w\parallel}$ and thus is located near the bottom of the
$\phi_{E\parallel}$-$v_{\parallel}$ diagram in Fig. 3c (the black region,
which represents the background ion concentration). Since the trajectories of
the deeply-trapped ions (including the green ion) cannot intersect the high-
PSD region, the phase space occupied by these ions (the resonance island
center) can only have lower densities due to the PSD conservation. The
shallowly-trapped ions, however, have trajectories intersecting the black
region (the red trajectories) so that the high-PSD solar wind ions can easily
access the outer edge of the resonance islands. Note that there are also
regions with low initial PSDs along the red trajectories, and therefore, it
would take longer than a Landau trapping period for the ions with higher and
lower PSDs to mix in phase space and produce rings with intermediately-high
PSDs (Fig. 3d). Here, the Landau-trapping period of $\sim$ 6s is indeed
shorter than the formation time ($\sim$30s) of phase-space rings in the
simulations.

To examine the scenario of phase-space ring formation via Landau trapping, we
launch two typical ions for their trajectories in the modeled field. The green
ion trajectory in Fig. 3e-f shows minor oscillations of $\phi_{E\parallel}$
and $v_{\parallel}$ around 180° and -100 km/s (approximately the parallel wave
velocity $v_{w\parallel}$), respectively. Their minor variations indicate that
the ion belongs to the deeply-trapped population. On the other hand, the red
ion experiences a larger variation of $\phi_{E\parallel}$ from $\sim$90° to
$\sim$270° (Fig. 3g), which indicates that it is trapped, albeit more
shallowly, within the potential well (otherwise, $\phi_{E\parallel}$ should
cover the full range from 0° to 360°). Moreover, its $v_{\parallel}$ changes
significantly from $\sim$-300km/s to $\sim$100km/s (Fig. 3h), which indicates
that solar-wind ions with higher PSDs can indeed be transported in phase space
along closed trajectories around the resonance island. Therefore, we conclude
that the formation of phase-space ion rings is associated with particle
trapping, serving as the observational identification of nonlinear Landau
resonance.

We next return to the simulated ion gyro-phase spectra (Fig. 2d-e), which
reproduce the phase-bunched features in observations (Fig. 1h-i). To confirm
that the phase-bunched features are caused by anomalous resonance, we show in
Fig. 4a the $v_{\parallel}-\zeta$ portrait of the ion phase-space
trajectories. Here, for simplicity, we assume field-aligned wave propagation,
with the effect of oblique propagation to be discussed later on. There are two
resonance islands occupying different $v_{\parallel}$ ranges, centered at
$\zeta$=0° (purple lines) and $\zeta$=180° (blue lines), respectively[16]. The
blue island centered at $\zeta$=180° corresponds to lower $v_{\parallel}$
values. Therefore, the nearby ion trajectories, including the closed (blue)
and the traversing trajectories (yellow), can access the background solar-wind
ions ($v_{\parallel}\sim-370$km/s), which leads to the concentration of high-
PSD ions around the $\zeta$=180° island. Fig. 4a also displays the observed
$v_{\parallel}-\zeta$ spectrum (averaged from 1625:33.7 to 1625:35.7), in
which the high-PSD ions are indeed concentrated around $\zeta$=180°. This
scenario is consistent with the superimposed trajectories, contributing to the
inclined stripes in observations (Fig. 1h-i) and simulations (Fig. 2d-e).

Figure 4:  Typical particle trajectories in the simulation. (a) The
$v_{\parallel}-\zeta$ spectrum, overplotted by ion trajectories. (b) $\zeta$
variation.

##  IV 4\. Discussion and Summary

The simulated ion trajectories in oblique waves could be more complicated than
expected from the $v_{\parallel}-\zeta$ diagram. Fig. 4b shows the $\zeta$
variations of a sample ion, which switches back and forth between traversing
(when $\zeta$ varies monotonically from 360° to 0°) and trapped trajectories
(when $\zeta$ oscillates around $\zeta$=0° or 180°) within either island. The
switch between different types of trajectories is usually attributed to field
variations[16, 2], but this is not the case here. Instead, the reason lies in
the existence of $E_{\parallel}$ (a property of oblique whistler waves), which
perturbs the parallel ion motion in the $v_{\parallel}-\zeta$ diagram and
consequently destabilizes the ion trajectories. As a result, the phase-bunched
stripes in Fig. 1i and 2e are not homogeneous, with minimum PSDs near
$\phi_{E\parallel}\sim$180° (see arrows in Fig. 1i and 2e) where the ions are
Landau trapped by $E_{\parallel}$ (see the white dashed lines in Fig. 2c,
corresponding to the low-PSD centers).

In other words, the Landau and anomalous resonances are largely coupled. This
coupling process, in analogy to the coupling between Landau and cyclotron
resonances [38, 39], is caused by the coexistence of the $q\bm{E}_{\parallel}$
and $q\bm{V}_{\perp}\times\bm{B}_{\perp}$ forces in the parallel direction,
and more importantly, the overlapped resonance islands in the $v_{\parallel}$
range between -300km/s and 100km/s. The coupling does not only affect the
anomalous resonance; the Landau resonance can also be perturbed by the
$q\bm{v}_{\perp}\times\bm{B}_{\perp}$ force, although in our case, the
$q\bm{v}_{\perp}\times\bm{B}_{\perp}$ force is much smaller than
$q\bm{E}_{\parallel}$ due to the ion concentration at $\zeta$=180°.
Consequently, the disturbances in Fig. 1g and Fig. 2c are much weaker. This
additional force, however, could still revise the particle trapping period to
expedite the phase-mixing process in the ring formation [1].

In summary, we demonstrate the simultaneous occurrence of Landau and anomalous
resonances between oblique whistler waves and solar wind ions. The two
resonant processes are strongly coupled through the overlapped resonance
islands, which is made possible by the coexistence of wave-associated electric
and Lorentz forces in the parallel direction. For anomalous resonance, the ion
trapping and traversing behavior are perturbed by the additional electric
force, which obscures the characteristic, phase-bunched features. The
diagnostic signatures for Landau resonance (phase-space rings in Fig. 1g and
2c), however, are perturbed only slightly by the wave-associated Lorentz force
due to the ion concentration near $\zeta$=180°. As the coupling strengthens,
more chaotic ion motion and more complicated features could be expected, which
may further revise the picture of foreshock wave-particle energy transfer.
This is beyond the scope of this paper, and we will carry out a more
systematic study to better understand the coupling process.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
grants 42174184. We are grateful to the MMS mission for the high-resolution
measurements. The MMS data are processed and analyzed using the IRFU-Matlab
package.

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in 1,…,0 See pages \x, of supplemental_material.pdf