# Turbulence properties and kinetic signatures of electron in Kelvin-Helmholtz
waves during a geomagnetic storm.

Harsha Gurram Department of Astronomy, University of Maryland College Park,
College Park, Maryland, USA.
NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.  Jason R. Shuster
Space Science Center, University of New Hampshire, Durham, New Hampshire, USA.
Li-Jen Chen  NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.
Matthew R. Argall Space Science Center, University of New Hampshire, Durham,
New Hampshire, USA.
Richard E. Denton  Department of Physics and Astronomy, Dartmouth College,
Hanover, New Hampshire, USA.  Rachel C. Rice  Department of Astronomy,
University of Maryland College Park, College Park, Maryland, USA.
NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.  Brandon L.
Burkholder  NASA Goddard Space Flight Center, Greenbelt, Maryland, USA.
University of Maryland Baltimore, Baltimore, Maryland, USA.
Daniel J. Gershman  NASA Goddard Space Flight Center, Greenbelt, Maryland,
USA.

###### Abstract

The Kelvin-Helmholtz instability (KHI), in its non-linear phase, plays a
significant role in transporting solar wind plasma into Earth’s magnetosphere.
This paper presents a comprehensive study of Magnetospheric Multiscale (MMS)
spacecraft encounter with KHI during a geomagnetic storm, focusing on
elucidating key turbulence properties and reconnection signatures observed at
the edges of KH vortices. The spectral slope for electric field stays
approximately constant for frequencies below the ion cyclotron frequency
($f<f_{ci}$) and exhibits a break around the lower hybrid frequency
($f_{lh}$), indicating wave activity. Furthermore, MMS observes a current
sheet accompanied by intense electron jets and features consistent with strong
guide-field asymmetric reconnection across the magnetopause. Substantial
agyrotropy (by a factor of $10$) in electron distribution functions is
observed in the reconnecting current sheet and at the edges of KH. Our
observation presents a multi-scale view into KH turbulence under strongly
driven conditions and into the dynamics occurring at electron dissipation
scales.

Magnetic Reconnection— Kelvin-Helmholtz waves –Turbulence — Magnetospheric
Multiscale (MMS) mission — Electron agyrotropy—Geomagnetic storm.

##  1 Introduction

The Kelvin-Helmholtz instability (KHI) is a classical instability that arises
at the interface between two fluids with non-zero relative velocity, and
occurs in various plasma environments, including at the edge of coronal mass
ejections in the solar corona (Foullon et al. (2011); Nykyri & Foullon (2013))
and in planetary magnetospheres (Kivelson & Chen (1995); Fairfield et al.
(2003); Delamere et al. (2013); Hasegawa et al. (2004)). At Earth’s
magnetopause, this instability can be triggered in the low-latitude
magnetopause boundary layer (LLBL) by the significant velocity shear (Fujimoto
et al. (1998); Hasegawa et al. (2004); Taylor et al. (2012); Haaland et al.
(2020)). The KH instability on the surface of the magnetopause begins as an
anti-sunward surface waves, which ’rolls up’ into a train of vortices in the
waves’ non-linear phase (Nakamura (2021); Kavosi & Raeder (2015)). These
vortices propagate towards the flanks of the magnetosphere and extend into the
tail region, playing a crucial role in facilitating the efficient entry of
collisionless solar wind plasma into the magnetosphere. KH waves disrupt the
frozen-in condition inherent in ideal magnetohydrodynamics (MHD) and play a
crucial role in transporting mass and momentum into the magnetosphere through
the kinetic process of magnetic reconnection within these vortices (Nykyri &
Otto (2001, 2004); Li et al. (2016); Ma et al. (2017); Nykyri et al. (2017)).

The first in-situ evidence of rolled-up KH vortices at the Earth’s flank
magnetopause was reported by Cluster spacecraft, confirming the presence of KH
instability under northward solar wind conditions (Hasegawa et al. (2009)).
These observations showed both cold solar wind particles ($<2$ keV) and hot
magnetospheric ions ($>5$ keV) within the vortices, suggest that KHI is the
primary mechanism for plasma transport across this boundary. The spacecraft
also encountered a current sheet of thickness a few times the ion-inertial
length at the trailing edge of the KH wave accompanied with strong Alfvénic
outflow jets parallel to the magnetosheath flow (Hasegawa et al. (2009)).
While the Cluster mission could resolve the vortices at ion scales, the
Magnetospheric Multiscale (MMS) mission, equipped with high-resolution
instrumentation, has offered an unprecedented opportunity to study KH waves at
electron scales, enabling examination of small-scale structures and dynamics
within the vortices, as well as investigation of micro-physical processes
responsible for particle energization and transport. MMS has provided direct
kinetic evidence of magnetic reconnection(Eriksson et al. (2016a, b); Li et
al. (2016)) including electron-only reconnection without ion coupling (Phan et
al. (2018)) in KH waves (Blasl et al. (2022)). Other KH-induced mechanisms
observed with MMS are mid-latitude reconnection (Vernisse et al. (2016)) and
turbulence (Stawarz et al. (2016)).

The KHI can drive turbulence in the magnetosheath region, acting as a large-
scale mechanism which initiates or reinforces a non-linear cascade (Hasegawa
et al. (2020)). Turbulence within KHI can induce reconnection, causing
magnetosheath electrons to traverse the magnetopause and magnetospheric
electrons to escape into the magnetosheath (Eriksson et al. (2016a); Li et al.
(2016)). Observations suggest the presence of Kelvin–Helmholtz waves on both
the dawn and dusk flank of the terrestrial magnetosphere (Otto & Fairfield
(2000); Fairfield et al. (2003); Hasegawa et al. (2004); Stawarz et al.
(2016). The first observations from the MMS of the KHI at the dusk-flank
magnetopause during southward interplanetary magnetic field were reported by
Blasl et al. (2022). The measurements show kinetic-scale electric field
fluctuations on the low-density side of the edges of waves due to lower-hybrid
drift instability (LHDI) excited by the density gradient at the edges of the
waves. This instability and fast reconnection disrupt the flow of non-linear
KH vortex, leading to a quick decay of the vortex structure possibly
explaining the difference in the observation probability of KH waves between
northward and southward IMF conditions (Nakamura et al. (2020, 2022); Blasl et
al. (2022)).

In this study we analyze KHI event elucidating the key turbulence properties
in KH waves during a storm by examining the wave power spectrum. Furthermore,
we will discuss the kinetic signatures of electrons near reconnecting current
sheets, present at the edges of the KH vortices. The article is organized as
follows: In Section 2, we provide an overview of the event under study. In
Section 3, we analyze magnetic and electric field spectra to understand
turbulence properties and energy transfer in KH. In Section 4, we present
observed signatures of reconnection, and report the key signature of electron
agyrotropy observed near reconnecting current sheet. Section 5 is dedicated to
quantifying this agyrotropy and offering insights into its origin. Finally, we
draw our conclusions in Section 6.

##  2 Overview of MMS observations on 14 April 2022

Figure 1: Overview of Kelvin-Helmholtz vortices encountered by MMS1 during a
geomagnetic storm on 14 April 2022. The panels show a) magnetic field
components, b) electron density, c) ion velocity components, d) electron
velocity components, e) ion and electron temperatures, f) omnidirectional
electron energy spectrum and g) omnidirectional ion energy spectrum. The
magenta box represents the reconnection current sheet discussed in Section 4.
An orbital plot of the MMS spacecraft in h) XZ and i) XY plane. All vectors
are in GSE coordinates.

During a geomagnetic storm on 14th April 2022, OMNI recorded a decrease in
sym-H (minimum of $-86$nT). MMS captured the IMF turning and magnetospheric
response from the equatorial dawn flank. MMS made a crossing into the day-side
magnetopause at around 10:15UT. From 11:32UT to 12:40UT MMS observes
quasi‐periodic high energy populations correspond with fluctuations in density
(Rice et al. (2024)). These quasi-periodic fluctuations have been identified
as KH waves at the dawn-flank of the magnetopause, exhibiting the
characteristics listed in Rice et al. (2022). Fig. 1 displays an overview of
the KH interval of interest observed by MMS1. During this period the IMF
pointed in the northward direction. Fig. 1a) exhibits quasi-periodic
fluctuations in the magnetic field component, which are correlated with the
fluctuations in electron density and electron temperature shown in Fig. 1b)
and e) respectively. The periodic observation of the magnetosheath and
magnetospheric regions is clearly evident in the density and temperatures as
MMS alternately encounters regions of plasma from the cold, dense
magnetosheath and the hot, tenuous magnetosphere.

To identify whether the observed fluctuations correspond to KH waves we
analyzed the data(not shown) in a LMN system obtained by applying minimum
variance analysis on electric field (MVA-E) (Rice et al. (2022)). The MVA-E
was performed on MMS1 data from 10:45 to 12:45 yielding a global coordinate
system given by: $\mathbf{L}_{{}_{\mbox{\scriptsize
GSM}}}=[0.5398,0.5815,0.6087]$, $\mathbf{M}_{{}_{\mbox{\scriptsize
GSM}}}=[0.6048,0.2351,-0.7609]$ and $\mathbf{N}_{{}_{\mbox{\scriptsize
GSM}}}=[0.5856,-0.7789,0.2247]$. Key features aiding in identifying KH waves
include semi-periodic fluctuations in magnetosphere and magnetosheath-like
plasma properties, such as densities, temperatures, bipolar fluctuations in
the normal component of the magnetic field ($B_{N}$) at the magnetopause, and
significant changes in tailward velocity. These fluctuations, typically lasted
3-7 minutes and accompanied by velocity shears, distinguish KH waves from
surface waves, with the observed sawtooth pattern and bipolar fluctuations in
the $B_{N}$ component serving as characteristic indicators of KHIs. To
differentiate from flux transfer events (FTEs), alignment of pressure minima
with $B_{N}$ is checked, where in KHIs, pressure is lowest in the vortex
center where $B_{N}$ is near $0$, whereas in FTEs, pressure increases when
$B_{N}$ approaches $0$, providing verifiable indicators for KHIs.

Fig. 1 shows MMS1 observations between 12:20:00 UT and 12:40:00UT when the
spacecraft encounters the upper boundary layer (Rice et al. (2024)). The
average MMS1 position is at $[1.6,-13.9,2.8]R_{\mbox{\scriptsize E}}$ in the
GSE system. Panels 1h) and 1i) display quasi-periodic perturbations in the
omnidirectional energy spectrum of ions and electrons indicating alternating
regions of plasma with energies typical of the magnetosheath and
magnetosphere, as well as mixed energies due to plasma mixing in the KH
vortex. The sharp increase in the magnetic field component $B_{z}$ from $20nT$
to $60nT$ corresponds to the edges of rolled up KH vortices or the sunward
facing edges (e.g., Otto & Fairfield (2000); Li et al. (2016); Kieokaew et al.
(2020)). These edges mark the transition between hot and tenuous
magnetospheric plasma to cold and dense magnetosheath-like plasma. Some of
these crossings exhibit a sharp magnetic field reversals (in $B_{y}$)
confirming the presence of current sheets at the edge of the vortices. One
such current sheet is highlighted within a magenta box in Fig. 1. It is
noteworthy that this specific region also exhibits reconnection signatures,
which we will delve into further in Section 4.

##  3 Power Spectral Analysis of KHI

Figure 2:  Fluctuations, Power spectra density (PSD) and local slope $a$ for
a) magnetic field, $\mathbf{B}$ and b) electric field, $\mathbf{E}$ for the
full KHI interval. Vertical dotted lines in grey, red, black and blue indicate
the frequencies $f_{ci}$, $f_{lh}$, $f_{pi}$ and $f_{ce}$ respectively.

During the investigated interval, the KHI is in its non-linear phase resulting
in the breakdown of vortices into smaller structures and initiating energy
cascading. Strong turbulence in the KHI is indicated by fluctuations in the
magnetic field with $|\delta B_{z}|/|B|\sim 0.8$. To analyze turbulence
properties, we examine the power spectrum of magnetic and electric
fluctuations. For the $B$ spectra, we use measurements from Fluxgate-
Searchcoil Merged (FSM) (Argall et al. (2018)) data product created by
combining data from the Fluxgate (Russell et al. (2016)) and Search Coil (Le
Contel et al. (2016)) Magnetometers. For the $E$ spectra the measurements from
the instrument Electric field Double probes (EDP) (Ergun et al. (2016);
Lindqvist et al. (2016)) is analyzed. Both instruments sample at a frequency
of $8192$ samples/second. The characteristic frequencies for this event are
ion cyclotron $f_{ci}\approx 1$Hz, lower hybrid $f_{LH}\approx 310$Hz, ion
plasma $f_{pi}\approx 1000$Hz and electron cyclotron $f_{ce}\approx 1500$Hz.

Fig. 2(a) and (b) shows the omni-directional magnetic and electric field
components and spectra for the KHI interval respectively. The local spectral
slope for the spectra was computed from power-law fit in the log–log plane.
The trace magnetic fluctuation spectrum follows a power law with the spectral
index of $-1.67$ for $f<f_{ci}$ before exhibiting steepening for $f>f_{ci}$.
This slope is close to Kolmogorov scaling with $-5/3$ slope. At the electron
scales the local spectra slope has a value $-2.69$ (close to $-8/3$) up to
$f_{lh}$. The trace electric field spectrum also follows a power-law and has a
spectral index of $-1.4$. Unlike the $B$ spectra, the $E$ spectral slope
flattens first at the electron scales and then exhibits steepening. The
flattening in the slope occurs between $f_{ci}$ and $f_{lh}$ around $8$Hz with
a slope of $-0.7$ and steepens beyond $f_{lh}$ with slope $-2.15$. For higher
frequencies the local slope increases due to flattening of the spectrum which
is not physical and thought to be due to aliasing of spin tone harmonics
(Koval & Szabo (2013)).

Both $E$ and $B$ spectra obey the theoretical fluid predictions and
observations of turbulence in magnetosheath in the low-frequency range
$f<f_{ci}$ (Stawarz et al. (2016); Chen et al. (2014)). The break in the
magnetic spectrum happens at $f_{ci}$ which is associated to length scales of
$d_{i}$. This can be related to the scale of reconnecting current sheets in
KH, which generally have a thickness $d_{i}$ (Leamon et al. (2000); Dmitruk et
al. (2004)). However, at these frequencies, the electric field spectra show a
constant spectral slope, suggesting that reconnection in KH vortex has reached
a steady state where the electric field is constant while the magnetic field
is still reconnecting. The electric field spectrum, however, exhibits a break
at frequencies closer to $f_{LH}$. This break indicates a flattening of the
spectrum, implying electric energy is not being released like the magnetic
energy leading to a buildup of electric fields and possible gain in electric
energy. The significant buildup in electric fields is dissipated by lower
hybrid waves possibly, as evidenced by the steepening of the electric spectra
beyond $f>f_{lh}$. During magnetic reconnection in a region of strong
turbulence in the Earth’s magnetotail (Ergun et al. (2020, 2022)) as well as
during KHI with reconnecting current sheets (Stawarz et al. (2016)) the $B$
spectra slope was $-3.0$ for $f>f_{ci}$ and $E$ spectra slopes were $-0.8$ for
$f_{ci}<f<f_{lh}$ and $-2.7\pm 0.2$ for $f>f_{lh}$. The storm event under
discussion exhibits shallower spectral slopes at $f>f_{ci}$, indicating slower
dissipation possibly due to the buildup of more electromagnetic energy and
fluctuations in KHI. Further investigation will be conducted to understand the
effects of geomagnetic storms on the nature of KH turbulence during storms and
their impact on the structure and evolution of KH turbulence and its
effectiveness in mixing and transport.

##  4 Reconnection signatures in the Kelvin-Helmholtz vortices

We investigated the full interval of 20min for evidence of current sheets with
reconnection signatures i.e. ion and electron jets along with field reversals.
The 20min burst data showed multiple current sheet crossings at the trailing
edge i.e. magnetosphere-magnetosheath edge of the KH vortices. This section
will focus on the current sheet observed at 12:28:48, highlighted with the
magenta box in Fig. 1, as it was accompanied by strong electron flow reversals
across the current sheet. This particular current sheet stands out in the
context of the overall event due to the pronounced anisotropy in temperature
observed within the inflow region. Additionally, our analysis reveals the
presence of a notable agyrotropy characterized by stretching and rotation in
the plane perpendicular to the magnetic field.

Figure 3: MMS1 burst observations in local LMN at 12:28:40-12:29:00 UT on 14
April 2022. The panels show a) magnetic field components, b) electron density
c), d), and e) electron velocity components $v_{L}$, $v_{M}$ and $v_{N}$, f)
parallel and perpendicular temperatures of electrons. The bottom three rows of
panels show the electron distribution functions at 5 different times
corresponding to the dashed lines panels in (a-e) in the i$1$-i$5$) $v_{\bot
1}-v_{\bot 2}$ plane, j$1$-j$5$) the $v_{||}-v_{\bot 1}$ plane and k$1$-k$5$)
$v_{||}-v_{\bot 2}$ plane. The electron distributions exhibit the anisotropy
in the parallel direction, hence we see a temperature anisotropy of about
$T_{e||}/T_{e\bot}\sim 3$. The eVDF at 12:28:45.100 exhibits agyrotropy,
wherein the eVDF has a significant elongation in the $v_{\bot 2}$ direction.
g) Cartoon showing the reconnection site on the KH vortex, h) The
reconstructed reconnection geometry using the method of Denton et al. (2022)
showing all the four spacecrafts. Between $t=51.1$ and $t=51.4$ the MMS2
spacecraft crosses the separatrix starting from the inflow to the exhaust.
This crossing is evident in the eVDF at $t=51.306$ wherein we can observe an
asymmetry in $v_{||}$, which indicates the mixing of magnetospheric and
magnetosheath electron populations typically observed at the separatrix. Here
$d_{sc}$ is the spacecraft separation. l1-l2) MMS2 eVDF at $t=51.306$ and
$51.606$ in $v_{||}-v_{\bot 2}$ plane.

Fig. 3a-f) shows 20 seconds of burst data of the magnetic field and the
electron velocity field components in the local LMN frame. The boundary normal
to the current sheet was estimated by applying minimum variance analysis on
the magnetic field (MVA-B) observed by MMS1. The guide-field $\mathbf{M}$
direction is the unit vector along of the cross product between $\mathbf{M}$
and the maximum variance direction of $\mathbf{B}$ and
$\mathbf{L}=\mathbf{M}\times\mathbf{N}$ completes the system. We performed
MVA-B for interval: 12:28:49 to 12:28:53 yielding the directions of LMN as:
$\mathbf{L}_{{}_{\mbox{\scriptsize GSE}}}=[-0.237,-0.823,0.516]$,
$\mathbf{M}_{{}_{\mbox{\scriptsize GSE}}}=[-0.215,-0.474,-0.854]$ and
$\mathbf{N}_{{}_{\mbox{\scriptsize GSE}}}=[0.947,-0.313,-0.0646]$. During this
20s interval, MMS-1 spans the boundary layer ($n_{1}=6cm^{-3}$) and
magnetosheath ($n_{2}=20cm^{-3}$) with an average $d_{i}=c/\omega_{pi}=73$km,
where c is the speed of the light and ion plasma frequency
$\omega_{pi}=(n_{1}e^{2}/m_{p}\epsilon_{0})$. The magnetic field showed a
rotation from $B_{L1}=-20$nT to $B_{L2}=10$nT. This rotation was accompanied
by a very weak (almost zero) $\mathbf{B}_{N}$ and a strong guide field
$\mathbf{B}_{M}$ component of $50$nT. This corresponds to a guide field of
$B_{M}/B_{L}=2.2$, supported by in-plane Hall electron currents. The current
sheet associated with this rotation of magnetic field lasts for 5 seconds.
During this interval, the spacecraft recorded an electron jet
$v_{eL1}=200$km/s and a ion jet of the order $v_{iL}=100$km/s (not shown). The
out-of-plane electron jets $v_{eM}$ experienced strong reversals along the
current sheet of about $|v_{eM}|=500$km/s.

The local LMN coordinate system reported above was utilized to determine the
reconnection geometry through polynomial reconstruction techniques developed
by Denton et al. (2022). Fig. 3h) show the reconstructed reconnection geometry
at time 12:28:51.1 and 12:28:51.4 UT respectively. From the reconstruction
results, it can be inferred that MMS2 encounters the separatrix around
12:28:51.1 UT and crosses into the exhaust at 12:28:51.4 UT. Both MMS2 and
MMS3 were observed to encounter the separatrix with a delay of $\Delta
t=300ms$. The separatrix crossing by MMS2 is also clearly evident in the
electron distributions obtained by the DES instrument. At $51.3$, the electron
distribution shows an asymmetry in $v_{||}$, where, $||$ refers to the
direction parallel to the magnetic field. This asymmetry in $v_{||}$ indicates
the mixing of two different electron populations, i.e in flowing magnetosheath
electrons with the $v_{||}>0$ and exhaust electron populations with
$v_{||}<0$. All four MMS spacecraft start in the inflow region on the
magnetosheath side, and MMS2 crosses the separatrix at $51.306$. This is
further evident by the strong temperature anisotropy observed in the
electrons. Fig. 3g) shows the cartoon where the reconnection X-line is on the
KH vortex, this corresponds to Type-I vortex induced reconnection(VIR). In
Type-I VIR, the in-plane magnetic field is initially anti parallel across the
velocity shear layer, and results in plasma mixing across the velocity shear
layer along the reconnected field lines (Li et al. (2016)). This reconnection
occurs when the magnetopause current layer is locally compressed by the vortex
flow resulting in the direct mixing of plasma across the shear layer.

The bottom three panels in Fig. 3 show the electron distribution functions
(eVDFs) at five different times in i1-i5) $v_{\bot 1}-v_{\bot 2}$ plane,
j1-j5) $v_{||}-v_{\bot 1}$ plane and k1-k5) $v_{||}-v_{\bot 2}$ plane for
MMS1, where $\bot 1$ refers to the
$(\mathbf{b}\times\mathbf{v})\times\mathbf{b}$ direction and $\bot 2$ refers
to the $(\mathbf{b}\times\mathbf{v})$ direction. These distributions
correspond to times 12:28:44.100, 12:28:45.100, 12:28:46.100, 12:28:49, and
12:28:53 UT from left to right marked as (1)-(5) in Fig. 3 respectively. The
eVDFs show a strong anisotropy in the parallel direction as shown in the
bottom two rows of Fig. 3 at all the marked times which is in agreement with
the reconstruction results that suggest the MMS1 is in the inflow region of
the reconnecting current sheet. The anisotropy in the electrons
$T_{e||}/T_{e\bot}$ is about 3. Additionally, the eVDFs exhibit agyrotropy in
the $v_{\bot 1}$-$v_{\bot 2}$ plane, in the form of elongation along $v_{\bot
2}$ direction as demonstrated by the eVDF at 12:28:46.100 UT. The eVDFs at
12:28:44.100 and 12:28:49 UT also exhibit agyrotropy, however unlike the eVDF
at 46.100 these eVDFs have elongation not purely along $v_{\bot 2}$, but at an
axis in $v_{\bot 1}-v_{\bot 2}$ plane. The three eVDFs exhibit a clockwise
rotation in the agyrotropy. At times 12:28:49 and 12:28:53 UT the eVDFs become
gyrotropic. During the 2 second interval 12:28:44.100 to 12:28:46.100 UT, when
the electrons are agyrotropic, an out-of-plane electron jet
$v_{M}=-200\mathrm{km/s}$ is observed. This jet is accompanied by a reversal
in the normal component of the electron velocity $v_{N}$ around 12:28:49
(marked (4) in Fig. 3). Furthermore, using a $B_{y}$-minimum feature we
estimated the structure velocity yielding $V_{{}_{\mbox{\scriptsize
nor}}}=203\,\mathrm{km/s}$. During this encounter, the spacecraft separation
is $d_{sc}=31km$. It was observed that the density ramp took place in about
$\sim 1s$, giving us a length scale for the density gradient of about $203km$.

Agyrotropic electrons have been previously observed by the MMS inside the
electron diffusion region of an asymmetric reconnection at the day-side
magnetopause(Egedal et al. (2016); Rager et al. (2018); Shuster et al.
(2019)). These agyrotropic signatures result from strong gradients which often
exist around the separatrices in asymmetric reconnection. However those VDFs
had a crescent-shaped distribution in the $v_{\bot 1}$ \- $v_{\bot 2}$ plane,
which arises due to spatial gradients in density and pressure. In contrast,
the electron velocity distribution functions observed near the separatrix of
reconnection at KH edges exhibit an ”elliptical” agyrotropic feature, a key
signature that has been not previously reported in KH studies. In the
following section, we will quantify and discuss the physical implications of
these agyrotropic signatures.

##  5 Signatures of electron agyrotropy in Kelvin Helmholtz waves

In regions where plasma properties like the density and the electron bulk
velocity vary sharply, we observe strong agyrotropic distributions. Previous
work has noted the relationship between agyrotropy and gradients at the scale
of the Larmor radius (Scudder et al. (2012, 2015)) primarily in reconnection
scenarios, simple model equilibria with density gradients that lead to the
gradients in the magnetized distribution function. In this section, we intend
to analyze the observed agyrotropic distributions qualitatively and
quantitatively. To quantify the agyrotropy in the electron VDFs we use the
following scalar parameter introduced by (Scudder & Daughton (2008)):

$A=2\frac{|P_{\bot e1}-P_{\bot e2}|}{P_{\bot e1}+P_{\bot e2}}$ (1)

Figure 4:  eVDFs exhibiting agyrotropy at the edges of KH vortices. a)
Magnetic field, b) density, c) parallel and perpendicular electron
temperatures, and d) electron agyrotropy. The dashed lines mark the times with
enhanced agyrotropy $A\sim 0.1$. e-i) eVDF’s in the $v_{\bot 1}-v_{\bot 2}$
plane at these marked times. We used the definition provided by Scudder &
Daughton (2008) to measure the electron agyrotropy as given by Eqn. 1. e1) and
f1) MMS1 measurements of ${\bf v}\cdot\nabla f_{e}$ at 12:20:56.192 and
12:28:46.066 UT. The absence of electron counts near $|v|\leq 2000$kms-1 in
these MMS distributions is due to the subtraction of low-energy photo-electron
contamination (Gershman et al. (2017)).

where $P_{\bot 1}$ and $P_{\bot 2}$ are pressures along $\bot 1$ and $\bot 2$
direction respectively. This agyrotropy parameter measures the deviations of
the VDFs in perpendicular velocity space from a circular shape, which is
characteristic of gyrotropic plasmas. A value of $A=0$ indicates gyrotropic
plasma, while an extreme value of 2 signifies highly non-gyrotropic plasma.
Fig. 4 d) shows plot of the agyrotropy parameter during the KHI event. The
plot shows spikes at the boundaries of KH waves, with an average agyrotropy
measure of approximately 0.1 at these edges, in contrast to an average value
of $0.015$ inside the vortices. This enhancement in agyrotropy by a factor of
$10$ at the edges in the electron distributions is evident in Fig. 4 e-i)
eVDFs which corresponds to times: 12:20:56.192, 12:28:45.066, 12:30:26.230,
12:33:44.380 and 12:38:30.000UT respectively.

In order to comprehend the origin of agyrotropy observed at the boundaries of
Kelvin-Helmholtz waves, our analysis will primarily focus on examining the
gradients in plasma parameters. It is important to note that the elliptical
nature of the distribution indicates that diamagnetic effects are unlikely to
be the driving force behind the observed electron agyrotropy. This is because
the kinetic signature of diamagnetic effects typically manifests as crescent-
shaped electron distributions, as demonstrated in previous studies (Egedal et
al. (2016); Rager et al. (2018); Shuster et al. (2019)). To gain a deeper
understanding of the kinetic origins of the elliptical agyrotropy, we refer to
recent observations from the Magnetospheric Multiscale (MMS) mission and
models that describe structures in terms of the electron Vlasov equation
(Shuster et al. (2021, 2023)). These works establish a fundamental connection
between different types of bulk plasma gradients and their kinetic
counterparts in the Vlasov equation. Specifically, they demonstrate that
quadrupolar velocity-space signatures in the spatial gradient term ${\bf
v}\cdot\nabla f_{e}$ are associated with spatial gradients in the bulk
velocity $\nabla U_{e}$ (Shuster et al. (2021)). For the specific KH event
under investigation in this study, electron moments data from the DES is
available for only three of the MMS spacecraft. As a result, it is not
possible to directly measure ${\bf v}\cdot\nabla f_{e}$. Nonetheless,
following the framework outlined in (Shuster et al. (2023)), we can utilize
single-spacecraft measurements of the velocity-space gradient term $({\bf
F}/m_{e})\cdot\nabla_{\bf v}f_{e}$ as a proxy for the spatial gradient term
for quasi-steady-state structures in kinetic equilibrium moving rapidly past
the spacecraft.

Fig. 4 e1) and f1) show characteristic examples of the Vlasov distribution
structures $({\bf F}/m_{e})\cdot\nabla_{\bf v}f_{e}$ at time 12:20:56.192 and
12:28:46.066 UT close to the times Fig. 4 e) and f) respectively, capturing
the occurrences of agyrotropy spikes. These 2D velocity-space slices show a
pronounced quadrupolar pattern in $({\bf F}/m_{e})\cdot\nabla_{\bf v}f_{e}$
that corresponds to and follows the rotating ‘propeller’-shaped elliptical
structure of the electron distributions. Decomposing the $({\bf
F}/m_{e})\cdot\nabla_{\bf v}f_{e}$ signature into the contribution from the
electric field $(-e/m_{e}){\bf E}\cdot\nabla_{\bf v}f_{e}$ and magnetic field
$(-e/m_{e})({\bf v}\times{\bf B})\cdot\nabla_{\bf v}f_{e}$ terms (not shown
here), we can interpret the quadrupolar pattern by recognizing the dominant
role played by the magnetic field term due to the substantial magnitude (a
factor of $\sim 3$) of $|{\bf B}|$. The term $(-{\bf v}\times{\bf
B})\cdot\nabla_{\bf v}f_{e}$ is equivalent to the gyrophase derivative term:
$B(\partial f_{e}/\partial\phi)$ (Shuster et al. (2023)). Consequently, the
elliptical agyrotropy observed in $f_{e}$ can be understood as an outcome of
this underlying kinetic structure: going counter-clockwise around a circle of
constant $|{\bf v}|$ in the $v_{\perp 1}$-$v_{\perp 2}$ plane, red regions
corresponding to $(\partial f_{e}/\partial\phi)>0$ result in an increase in
phase space density, while blue regions with $(\partial f_{e}/\partial\phi)<0$
indicate a decrease in phase space density. The presence of quadrupolar
signatures in $({\bf F}/m_{e})\cdot\nabla_{\bf v}f_{e}$ imply the existence of
electron-spatial-scale gradients in the electron bulk velocity, consistent
with previous findings. This is a reasonable inference since the overall KH
structure arises from larger ion-scale vortices, making it plausible for
smaller-scale embedded bulk velocity shears to develop at these electron
scales, which can now be probed by MMS instrumentation.

##  6 Summary and Discussions

In this paper, we explore the key physics underlying turbulence and
reconnection within KH vortices during a geomagnetic storm, utilizing
observations measured by MMS. Our analysis shows the presence of power-law
spectra in both electric and magnetic field data, indicating the occurrence of
turbulence. Notably, while the magnetic field power spectra demonstrate
expected changes in spectral slope near the ion gyrofrequency, the electric
field spectra do not exhibit a break at the ion cyclotron frequency; instead,
a break is observed near the lower hybrid frequency. Furthermore, we observe
shallower spectral slopes compared to turbulence in magnetotail region,
suggesting a slower dissipation to small scales, possibly attributable to the
storm conditions. The storm conditions may lead to increased energy buildup,
delaying dissipation to smaller scales. We are currently engaged in further
research to comprehensively understand turbulence generation within KH waves
under both storm and non-storm conditions.

During this event, the MMS spacecraft observed a current sheet located
approximately $18d_{e}$($\sim 0.6d_{sc}$) away from an electron diffusion
region, accompanied by intense electron jets. This aligns with features
associated with strong guide-field asymmetric reconnection occurring across
the magnetopause. Alongside a strong anisotropy ($T_{||}>T_{\bot}$) in the
inflow region the electrons show deviations from gyrotropic behavior near the
reconnecting current sheet. This agyrotropy was observed at the edges of the
KH vortices, where pronounced magnetic gradients are prominent. The measured
agyrotropy was approximately $0.1$. This agyrotropy manifests as an elliptical
structure in electron velocity distribution functions, distinct from the
crescent distributions extensively observed in MMS data. While the crescent
shaped distributions arise from density gradients, this agyrotropy arises from
the gradients in electron bulk flow velocity that arise possibly due to the
intense storm-time conditions or the large velocity shears at electron scales
inside the vortices. Further examination of other KH encounters by MMS
revealed similar agyrotropy in two additional events, one occurring during a
storm and the other under non-storm conditions. Consequently, the source of
this elliptical agyrotropy remains unanswered within the realm of KH studies,
prompting further investigation to determine the magnitude of electron-scale
gradients required for agyrotropy manifestation within reconnecting current
sheets, while also exploring the influence of these reconnecting current
sheets on KH wave power spectra.

We extend our appreciation to the MMS team for providing the data used in this
study, sourced from the MMS Science Data Center
(https://lasp.colorado.edu/mms/sdc/). MRA was supported by NASA grant
80NSSC18K1359.

## References

  * Argall et al. (2018) Argall, M. R., Fischer, D., Contel, O. L., et al. 2018, The Fluxgate-Searchcoil Merged (FSM) Magnetic Field Data Product for MMS.  https://arxiv.org/abs/1809.07388
  * Blasl et al. (2022) Blasl, K. A., Nakamura, T. K. M., Plaschke, F., et al. 2022, Physics of Plasmas, 29, doi: 10.1063/5.0067370
  * Chen et al. (2014) Chen, C. H. K., Leung, L., Boldyrev, S., Maruca, B. A., & Bale, S. D. 2014, Geophysical Research Letters, 41, 8081, doi: https://doi.org/10.1002/2014GL062009
  * Delamere et al. (2013) Delamere, P. A., Wilson, R. J., Eriksson, S., & Bagenal, F. 2013, Journal of Geophysical Research: Space Physics, 118, 393, doi: https://doi.org/10.1029/2012JA018197
  * Denton et al. (2022) Denton, R. E., Liu, Y.-H., Hasegawa, H., et al. 2022, Journal of Geophysical Research: Space Physics, 127, e2022JA030512, doi: https://doi.org/10.1029/2022JA030512
  * Dmitruk et al. (2004) Dmitruk, P., Matthaeus, W. H., & Seenu, N. 2004, The Astrophysical Journal, 617, 667, doi: 10.1086/425301
  * Egedal et al. (2016) Egedal, J., Le, A., Daughton, W., et al. 2016, Phys. Rev. Lett., 117, 185101, doi: 10.1103/PhysRevLett.117.185101
  * Ergun et al. (2016) Ergun, R. E., Tucker, S., Westfall, J., et al. 2016, Space Sci. Rev., 199, 167, doi: 10.1007/s11214-014-0115-x
  * Ergun et al. (2020) Ergun, R. E., Ahmadi, N., Kromyda, L., et al. 2020, ApJ, 898, 154, doi: 10.3847/1538-4357/ab9ab6
  * Ergun et al. (2022) Ergun, R. E., Pathak, N., Usanova, M. E., et al. 2022, ApJ, 935, L8, doi: 10.3847/2041-8213/ac81d4
  * Eriksson et al. (2016a) Eriksson, S., Wilder, F., Ergun, R., et al. 2016a, Physical Review Letters, 117, 015001, doi: 10.1103/PhysRevLett.117.015001
  * Eriksson et al. (2016b) Eriksson, S., Lavraud, B., Wilder, F. D., et al. 2016b, Geophysical Research Letters, 43, 5606, doi: 10.1002/2016GL068783
  * Fairfield et al. (2003) Fairfield, D. H., Farrugia, C. J., Mukai, T., Nagai, T., & Fedorov, A. 2003, Journal of Geophysical Research: Space Physics, 108, doi: https://doi.org/10.1029/2003JA010134
  * Foullon et al. (2011) Foullon, C., Verwichte, E., Nakariakov, V. M., Nykyri, K., & Farrugia, C. J. 2011, The Astrophysical Journal Letters, 729, L8, doi: 10.1088/2041-8205/729/1/L8
  * Fujimoto et al. (1998) Fujimoto, M., Terasawa, T., & Mukai, T. 1998, The Low-Latitude Boundary Layer in the Tail-Flanks (American Geophysical Union (AGU)), 33–44, doi: https://doi.org/10.1029/GM105p0033
  * Gershman et al. (2017) Gershman, D. J., Avanov, L. A., Boardsen, S. A., et al. 2017, Journal of Geophysical Research: Space Physics, 122, 11,548, doi: https://doi.org/10.1002/2017JA024518
  * Haaland et al. (2020) Haaland, S., Paschmann, G., Øieroset, M., et al. 2020, Journal of Geophysical Research: Space Physics, 125, e2019JA027623, doi: https://doi.org/10.1029/2019JA027623
  * Hasegawa et al. (2004) Hasegawa, H., Fujimoto, M., Phan, T.-D., et al. 2004, Nature, 430, 755, doi: 10.1038/nature02799
  * Hasegawa et al. (2009) Hasegawa, H., Retinò, A., Vaivads, A., et al. 2009, Journal of Geophysical Research: Space Physics, 114, doi: 10.1029/2009JA014042
  * Hasegawa et al. (2020) Hasegawa, H., Nakamura, T. K. M., Gershman, D. J., et al. 2020, Journal of Geophysical Research: Space Physics, 125, e2019JA027595, doi: https://doi.org/10.1029/2019JA027595
  * Kavosi & Raeder (2015) Kavosi, S., & Raeder, J. 2015, Nature Communications, 6, 7019, doi: 10.1038/ncomms8019
  * Kieokaew et al. (2020) Kieokaew, R., Lavraud, B., Foullon, C., et al. 2020, Journal of Geophysical Research: Space Physics, 125, e2019JA027527, doi: 10.1029/2019JA027527
  * Kivelson & Chen (1995) Kivelson, M. G., & Chen, S.-H. 1995, The Magnetopause: Surface Waves and Instabilities and their Possible Dynamical Consequences (American Geophysical Union (AGU)), 257–268, doi: https://doi.org/10.1029/GM090p0257
  * Koval & Szabo (2013) Koval, A., & Szabo, A. 2013, in American Institute of Physics Conference Series, Vol. 1539, Solar Wind 13, ed. G. P. Zank, J. Borovsky, R. Bruno, J. Cirtain, S. Cranmer, H. Elliott, J. Giacalone, W. Gonzalez, G. Li, E. Marsch, E. Moebius, N. Pogorelov, J. Spann, & O. Verkhoglyadova, 211–214, doi: 10.1063/1.4811025
  * Le Contel et al. (2016) Le Contel, O., Leroy, P., Roux, A., et al. 2016, Space Sci. Rev., 199, 257, doi: 10.1007/s11214-014-0096-9
  * Leamon et al. (2000) Leamon, R. J., Matthaeus, W. H., Smith, C. W., et al. 2000, The Astrophysical Journal, 537, 1054, doi: 10.1086/309059
  * Li et al. (2016) Li, W., André, M., Khotyaintsev, Y. V., et al. 2016, Geophysical Research Letters, 43, 5635, doi: 10.1002/2016GL069192
  * Lindqvist et al. (2016) Lindqvist, P. A., Olsson, G., Torbert, R. B., et al. 2016, Space Sci. Rev., 199, 137, doi: 10.1007/s11214-014-0116-9
  * Ma et al. (2017) Ma, X., Delamere, P., Otto, A., & Burkholder, B. 2017, Journal of Geophysical Research: Space Physics, 122, 10,382, doi: https://doi.org/10.1002/2017JA024394
  * Nakamura (2021) Nakamura, T. K. M. 2021, The Earth’s Low-Latitude Boundary Layer (American Geophysical Union (AGU)), 177–191, doi: https://doi.org/10.1002/9781119815624.ch12
  * Nakamura et al. (2020) Nakamura, T. K. M., Plaschke, F., Hasegawa, H., et al. 2020, Geophysical Research Letters, 47, e2020GL087574, doi: https://doi.org/10.1029/2020GL087574
  * Nakamura et al. (2022) Nakamura, T. K. M., Blasl, K. A., Hasegawa, H., et al. 2022, Physics of Plasmas, 29, 012901, doi: 10.1063/5.0067391
  * Nykyri & Foullon (2013) Nykyri, K., & Foullon, C. 2013, Geophysical Research Letters, 40, 4154, doi: https://doi.org/10.1002/grl.50807
  * Nykyri et al. (2017) Nykyri, K., Ma, X., Dimmock, A., et al. 2017, Journal of Geophysical Research: Space Physics, 122, 9489, doi: https://doi.org/10.1002/2017JA024374
  * Nykyri & Otto (2001) Nykyri, K., & Otto, A. 2001, Geophysical Research Letters, 28, 3565, doi: https://doi.org/10.1029/2001GL013239
  * Nykyri & Otto (2004) —. 2004, Annales Geophysicae, 22, 935.  https://hal.science/hal-00317273
  * Otto & Fairfield (2000) Otto, A., & Fairfield, D. H. 2000, J. Geophys. Res., 105, 21,175, doi: 10.1029/1999JA000312
  * Phan et al. (2018) Phan, T. D., Eastwood, J. P., Shay, M. A., et al. 2018, Nature, 557, 202, doi: 10.1038/s41586-018-0091-5
  * Rager et al. (2018) Rager, A. C., Dorelli, J. C., Gershman, D. J., et al. 2018, Geophysical Research Letters, 45, 578, doi: https://doi.org/10.1002/2017GL076260
  * Rice et al. (2022) Rice, R. C., Nykyri, K., Ma, X., & Burkholder, B. L. 2022, Journal of Geophysical Research: Space Physics, 127, e2021JA029685, doi: 10.1029/2021JA029685
  * Rice et al. (2024) Rice, R. C., Chen, L.-J., Gershman, D., et al. 2024, Geophysical Research Letters, 51, e2023GL106600, doi: https://doi.org/10.1029/2023GL106600
  * Russell et al. (2016) Russell, C. T., Anderson, B. J., Baumjohann, W., et al. 2016, Space Sci. Rev., 199, 189, doi: 10.1007/s11214-014-0057-3
  * Scudder & Daughton (2008) Scudder, J., & Daughton, W. 2008, Journal of Geophysical Research: Space Physics, 113, doi: 10.1029/2008JA013035
  * Scudder et al. (2012) Scudder, J. D., Holdaway, R. D., Daughton, W. S., et al. 2012, Phys. Rev. Lett., 108, 225005, doi: 10.1103/PhysRevLett.108.225005
  * Scudder et al. (2015) Scudder, J. D., Karimabadi, H., Daughton, W., & Roytershteyn, V. 2015, Physics of Plasmas, 22, 101204, doi: 10.1063/1.4932332
  * Shuster et al. (2019) Shuster, J. R., Gershman, D. J., Chen, L.-J., et al. 2019, Geophysical Research Letters, 46, 7862, doi: https://doi.org/10.1029/2019GL083549
  * Shuster et al. (2021) Shuster, J. R., Gershman, D. J., Dorelli, J. C., et al. 2021, Nature Physics, 17, 1056, doi: 10.1038/s41567-021-01280-6
  * Shuster et al. (2023) Shuster, J. R., Gershman, D. J., Giles, B. L., et al. 2023, Journal of Geophysical Research: Space Physics, 128, e2022JA030949, doi: https://doi.org/10.1029/2022JA030949
  * Stawarz et al. (2016) Stawarz, J. E., Eriksson, S., Wilder, F. D., et al. 2016, Journal of Geophysical Research: Space Physics, 121, 11,021, doi: https://doi.org/10.1002/2016JA023458
  * Taylor et al. (2012) Taylor, M. G. G. T., Hasegawa, H., Lavraud, B., et al. 2012, Annales Geophysicae, 30, 1025, doi: 10.5194/angeo-30-1025-2012
  * Vernisse et al. (2016) Vernisse, Y., Lavraud, B., Eriksson, S., et al. 2016, Journal of Geophysical Research: Space Physics, 121, 9926, doi: https://doi.org/10.1002/2016JA023051