# Possible coexistence of kinetic Alfvén and ion Bernstein modes in sub-ion
scale compressive turbulence in the solar wind

Owen Wyn Roberts<EMAIL_ADDRESS>Space Research Institute, Austrian
Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria  Daniel Verscharen
Mullard Space Science Laboratory, University College London, Dorking RH5 6NT,
UK and
Space Science Center, University of New Hampshire, 8 College Road, Durham, NH
03824, USA.  Yasuhito Narita  Space Research Institute, Austrian Academy of
Sciences, Schmiedlstrasse 6,8042 Graz, Austria  Rumi Nakamura  Space Research
Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,8042 Graz, Austria
Zoltán Vörös  Space Research Institute, Austrian Academy of Sciences,
Schmiedlstrasse 6,8042 Graz, Austria and
Geodetic and Geophysical Institute, Research Centre for Astronomy and Earth
Sciences (RCAES), Sopron, Hungary  Ferdinand Plaschke  Space Research
Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,8042 Graz, Austria

###### Abstract

We investigate compressive turbulence at sub-ion scales with measurements from
the Magnetospheric MultiScale Mission. The tetrahedral configuration and high
time resolution density data obtained by calibrating spacecraft potential
allow an investigation of the turbulent density fluctuations in the solar wind
and their three-dimensional structure in the sub-ion range. The wave-vector
associated with the highest energy density at each spacecraft frequency is
obtained by application of the Multi-point signal resonator technique to the
four-point density data. The fluctuations show a strong wave-vector anisotropy
$k_{\perp}\gg k_{\parallel}$ where the parallel and perpendicular symbols are
with respect to the mean magnetic field direction. The plasma frame
frequencies show two populations, one below the proton cyclotron frequency
$\omega<\Omega_{ci}$ consistent with kinetic Alfvén wave (KAW) turbulence. The
second component has higher frequencies $\omega>\Omega_{ci}$ consistent with
ion Bernstein wave (IBW) turbulence. Alternatively, these fluctuations may
constitute KAWs that have undergone multiple wave-wave interactions causing a
broadening in the plasma frame frequencies. The scale-dependent kurtosis in
this wave-vector region shows a reduction in intermittency at the small scales
which can also be explained by the presence of wave activity. Our results
suggest that small-scale turbulence exhibits linear-wave properties of kinetic
Alfvén and possibly ion-Bernstein/magnetosonic waves. Based on our results, we
speculate that these waves may play a role in describing the observed
reduction in intermittency at sub ion scales.

††preprint: APS/123-QED

_Introduction:_ Plasma turbulence is an inherently multi-scale, three-
dimensional phenomenon [1, 2, 3]. The solar wind is an easily accessible
turbulent plasma, and its temperature decreases slower with heliocentric
distance than expected for an adiabatically expanding gas [4]. The dissipation
of plasma turbulence is likely to play a role in the observed plasma heating
of the solar wind. It is still a matter of ongoing research as to what degree
the turbulent fluctuations are wave-like or rather characteristic of
intermittent coherent structures such as vortices or current sheets.

In the solar wind, turbulent fluctuations exist from $\sim 10^{6}$ km (termed
the correlation scale, the size of the largest eddie [2]) to electron scales
at $\sim 1$km. At scales smaller than the correlation scale fluid-like
nonlinear interactions between fluctuations cascade energy to smaller scales
and the power spectral density (PSD) of magnetic and density fluctuations at
1a.u. have a Kolmogorov like spectral index of -5/3 [1]. In this region, the
magnetic fluctuations are dominated by the transverse components. However,
there is a non-negligible component of energy in compressive fluctuations [5,
6]. At scales near the proton characteristic scales, a break is observed in
the spectrum of magnetic field fluctuations [7], and a steepening to a
spectral index of $\sim-2.6$ [8]. This region is termed the sub-ion range,
where plasma waves are dispersive or dissipation heats the plasma. In this
range, the relative power of the compressive component of the magnetic-field
fluctuations increases, eventually becoming comparable to the transverse
components [9].

A variety of linear waves can exist in a plasma, and when their frequencies
reach the characteristic scales of the particles then wave-particle
interactions can occur, transferring energy from the wave to the plasma [10].
At ion inertial scales the turbulence is thought to contain both kinetic
Alfvén waves, and slow waves [5, 11]. At smaller scales, the kinetic slow
waves are damped strongly leaving only kinetic Alfvén waves which become
increasingly compressible at sub-ion scales [12, 13]. Turbulence is also
characterized by intermittency [14]. Observations and numerical simulations
have shown that plasma heating is often located near intermittent coherent
structures [15]. Coherent structures exhibit multi-scale phase coherence [e.g.
16, 17, 18, 19, 20]. This is different to a plane wave which has phase
coherence in time over the wave-packet’s duration and is a single scale
phenomena [e.g. 21, 22]. The level of intermittency is known to increase as
the turbulent cascade progresses to smaller scales [23], however, the nature
of intermittency in the sub-ion range is unclear. In the Earth’s magnetosheath
[24] and the solar wind [25] observations show that the increase in the scale-
dependent kurtosis persists into the sub-ion range. However, other
observations in the solar wind [9] show this region is characterized by a non-
intermittent monofractal scaling, and the scale-dependent kurtosis of
fluctuations plateaus or even decreases [24]. The reason is unclear but it has
been suggested that high-frequency waves may have a randomizing effect on the
increments [26], similar to how Alfvén waves at fluid scales can cause the
transverse components to be less intermittent [27].

The nature of the fluctuations i.e. wave-like or structure-like has been a
topic of spirited debate in the plasma turbulence community. The exact
relationships between the two different phenomena and their interactions with
one another remain unclear. It is possible that both types of structures
coexist [e.g. 28], or that waves can generate structures [29] increasing
intermittency, or reduce it [27, 30]. Furthermore, structures can also
generate waves themselves [31]. It is also often hypothesized that even in the
highly nonlinear regime the linear physics remain important for the turbulence
dynamics [32, 33, 34].

The goal of this paper is to test the hypothesis that waves destroy the
intermittency in the sub-ion range. The problem is tackled through a
combination of wave-based analysis methods to identify and classify wave-like
fluctuations and increment based methods that measure the intermittency. Both
of these goals require data with high time resolution and sensitivity.
Furthermore, to determine the wave-vectors multiple spacecraft are required,
with distances of a few kilometers between them. This makes the data from the
Magnetospheric MultiScale Mission (MMS) the only data set suitable to explore
the problems described.

_Data/Methods:_ On December 6th 2016 at 11:37:34UT the MMS [35] spacecraft
record an interval of slow solar wind in burst mode until 11:44:04UT. During
this time the four spacecraft are in a regular tetrahedral formation ensuring
homogeneous spatial coverage with inter-spacecraft distances $\sim 6$km. The
mean values are as follows; the proton bulk speed is $350$km s-1, the electron
density is $16.9$ cm-3, the magnetic field strength is 7.6 nT, the ion and
electron plasma $\beta$ (ratio of thermal to magnetic pressure) values are 1.7
and 1.1 respectively. It has been reported that the Fast Plasma
Investigation’s Dual Ion Spectrometer (FPI-DIS) [36] may not have reliable
measurements of the ion temperatures in the solar wind [37]. The value of the
ion $\beta$ using the mean temperature from OMNI data [38] is 0.5. Figure 1
shows the measured data from the MMS, with the magnetic field being measured
by the fluxgate magnetometer (FGM) [39] at a sampling rate of 128Hz in the
Geocentric Solar Ecliptic (GSE) coordinate system. In this coordinate system x
points from the Earth towards the Sun and z points to the ecliptic north. The
electron density is measured using two different methods, the first being a
direct measurement from the FPI’s Dual Electron Spectrometers (FPI-DES) [36]
which have a sampling rate of 33Hz. The other method is to use the spacecraft
potential which is measured with a sampling rate 8.192kHz from the spin plane
double probes (SDP) [40]. We calibrate the spacecraft potential measurement
using lower resolution electron density from FPI to give a higher time
resolution measurement of the electron density [41, 5]. The procedure to infer
density from the spececraft potential has been used by several authors using
MMS data [e.g. 42, 43, 44, 45]. A detailed description of the application of
this method in the solar wind with MMS is available in [47].

The spacecraft potential is governed by the currents to and from the
spacecraft. These include, the electron thermal current to the spacecraft, the
photoelectron current from the spacecraft and several other currents. In a
cold, sparse plasma such as the solar wind the contributions of other currents
are typically negligible and the photoelectron current and the electron
thermal current are approximately equal and oppositely directed. The electron
thermal current can be estimated from the plasma measurements,

$I_{e}=-A_{\text{spac}}qn_{e}\sqrt{\frac{k_{B}T_{e}}{2m_{e}\pi}}\left(1+\frac{qV_{sc}}{k_{B}T_{e}}\right)$
(1)

Where $A_{spac}$ is the spacecraft area, $V_{sc}$ is the spacecraft potential,
$q$ is the fundamental charge, $m_{e}$ is the electron mass, $T_{e}=11.0$eV is
the measured electron temperature from FPI-DES for this interval and $k_{B}$
is the Boltzmann constant. The photoelectron current is modeled as an
exponential function;

$I_{\text{ph}}=I_{\text{ph0}}\exp{(-V_{sc}/V_{0})},$ (2)

where $I_{\text{ph0}}$ and $V_{0}$ are the photoelectron currents and
potentials obtained by fitting the electron thermal current from FPI-DES to
the spacecraft potential [e.g 41]. A table of the constants used for
calibrating this interval are given in Tab 1.

Table 1: Table showing the photoelectron parameters used for converting the spacecraft potential into a density |  $I_{ph0}$ ($\mu$ A) |  $V_{0}$ (V)
---|---|---
MMS1 | 301.5 | 2.0
MMS2 | 474.2 | 1.7
MMS3 | 375.9 | 1.9
MMS4 | 219.9 | 2.1

By equating Eq 2 and 1 and solving for the electron density we obtain;

$n_{e,SC}=\frac{1}{qA_{\text{spac}}}\sqrt{\left(\frac{2\pi
m_{e}}{k_{B}T_{e}}\right)}\left(1+\frac{qV_{sc}}{k_{B}T_{e}}\right)^{-1}I_{ph}$
(3)

The calibration procedure and the methodology for removing the spin is
discussed in more detail in [5, 47]. A comparison of the electron density
derived from the spacecraft potential and the direct measurement from the FPI-
DES is shown in Fig 1. The effects of charging timescales can be neglected as
we consider frequencies much lower than the charging timescales which are of
order a few kilo Hertz [e.g. 48] in the solar wind. For an estimation of the
timescale we assume the photoelectron flux is constant and therefore any
changes in the spacecraft potential will be due to the changes in the ambient
plasma. The photoelectron flux is dependent on the UV flux from the Sun,
primarily the Lyman Alpha emission [e.g. 49]. Which we expect to be constant
during the short interval studied here. The photoelectron emission can vary
due to strong electric fields ($\gtrsim 40$mV/m) [50, 43, 45], however, such
large electric fields are associated with magnetopause crossings, not quiet
solar wind. Therefore this assumption is justified. Following the calculation
of Chen  _et al._ [48] the charging timescale in the solar wind can be
approximated by $\delta t=(CV_{0})/I_{e}$, where $C$ is the total capacitance
of the spacecraft, which is estimated to be 2nF from the sum of the
capacitances of each component of the spacecraft [43], and $I_{e}$ is the
electron photocurrent which in the solar wind is typically of the order of
$30-40\mu$ A. This corresponds to a frequency of $\sim 7$kHz. An estimate of
the charging time scale of the MMS spacecraft in the solar wind using a dust
impact found that it was of the order of micro seconds [46]. Both of these
estimates correspond to frequencies that are much larger than the Nyquist
frequency of the Burst mode data and much larger than the instrumental noise
which occurs near 40Hz [47]. Therefore, the charging timescale will not affect
our results.

Figure 1: Measured time series from MMS from top to bottom. The magnetic field
fluctuations from FGM, the electron density from FPI-DES. The spacecraft
potential derived density from SDP.

To obtain spatial information about a plasma from single point measurements
Taylor’s hypothesis is often applied. This assumes the fluctuations evolve on
a timescale longer than advection time over the measurement point so that the
spacecraft sees a one-dimensional cut through the plasma. Should fluctuations
be dispersive this hypothesis breaks down, so that multi-point data analysis
techniques become necessary. To measure the wave-vector anisotropy and the
plasma frame frequencies of the fluctuations we apply the Multi-Point Signal
Resonator technique (MSR) [51]. This method uses data from four spacecraft and
filters the signal for a given spacecraft frame frequency and wave-vector
using the phase delay between spacecraft. This method assumes weak
stationarity of the time series and that it can be mathematically described as
a superposition of plane waves with random phases and a small component of
incoherent noise. However, crucially, Taylor’s hypothesis is not required.

The MSR technique estimates the wave power in four-dimensional space
$P(\omega_{sc},\mathbf{k})$ for a spacecraft frame frequency $\omega_{sc}$ and
a wave-vector $\mathbf{k}$. The peak in $P(\omega_{sc},\mathbf{k})$ can be
used to find the wave-vector associated with the highest energy density at a
given spacecraft frame frequency. This wave-vector is Doppler shifted to the
plasma frame via $\omega_{pla}=\omega_{sc}-\mathbf{k}\cdot\mathbf{v_{sw}}$
where $\mathbf{v_{sw}}$ denotes the mean ion bulk velocity. The
$\omega_{pla}$-$k$ relation and the angle between $\mathbf{k}$ and the mean
magnetic field direction can then be compared to linear solutions of the
Maxwell-Vlasov equation.

The accessible range of scales is determined by the inter-spacecraft
separations $d_{sc}$, where a maximum wave-number is given as
$k_{\text{max}}=\pi/\langle d_{sc}\rangle$ ; the Nyquist wave-number. This
corresponds to an upper limit of
$f_{\text{max}}=k_{\text{max}}(v_{sw}-v_{ph})/{2\pi}=(v_{sw}-v_{ph})/2\langle
d_{sc}\rangle\sim 24$Hz, where $v_{ph}=\textbf{Max}(c_{s},v_{A})$. For this
interval the sound speed $c_{s}=58$ km s-1 exceeds the Alfvén speed $v_{A}=38$
km s-1. The minimum wave-number and frequency is selected based on resolving
the wave-vector with an accuracy of $10\%$ which we set to
$k_{\text{min}}=k_{\text{max}}/25$ [52] for a plane wave. This corresponds to
a minimum frequency of $f_{\text{min}}\sim 1$Hz. We use the limit of
$f_{\text{min}}\sim 2$Hz as some of the recovered associated wavenumbers with
frequencies below 2Hz have large relative errors. We note that the magnetic
field measurement on MMS is not sufficiently sensitive to be used for the
majority of the frequency range in the solar wind and becomes noisy near
$f_{sc}\gtrsim 5$Hz, whereas the density data become noisy at $f_{sc}\gtrsim
40$Hz and have the necessary time resolution and sensitivity to cover the
range [$f_{\text{min}},f_{\text{max}}$]. Before inputting the data into the
MSR method the density data from all spacecraft are resampled onto the same
timeline, at a reduced rate of 5378.31Hz giving $2^{21}$ data points which are
split into 128 windows of 16384 data points which overlap by $50\%$ giving a
total of 256 windows. There are some instrumental effects in the spacecraft
potential data at 13-14 Hz which are discarded from our study.

Figure 2: (a) Fourier Power spectral densities (PSD) of the trace magnetic
fluctuations (b) PSD of the electron density fluctuations from FPI-DES. (c)
PSD of the magnetic field magnitude fluctuations (d) PSD of the electron
density fluctuations from the spacecraft potential (e) The ratio of the PSDs
of magnetic field magnitude fluctuations to the trace fluctuations. (f) The
cross-correlation spectra of $B$ and $n_{e}$. (g) the ratio of normalized
electron density and perpendicular magnetic fluctuations (h) the magnetic
helicity spectra. In figures (e-h) the black lines denote the quantities
obtained from Fourier analysis while the thick red curves denote the
estimation of the quantities based on wavelet analysis. The green and dark
green areas denote [$f_{\text{min}},f_{\text{max}}$]. The dark green area
denotes the region where FGM reaches the noise floor but where the density
measurement remains viable. The second y-axis in (e-h) shows the plasma frame
frequencies from the MSR technique which are overplotted for comparison with
the spectral estimates.

_Results Spectral Estimators:_ Figure 2(a-d) shows the Fourier power spectra
of the magnetic and density fluctuations. The dotted lines denote the Taylor
shifted gyroradius $f_{\rho p,e}=v_{sw}/2\pi\rho_{p,e}$ and inertial length
,$f_{dp,e}=v_{sw}/2\pi d_{p,e}$ for protons and electrons and
$f_{\rho_{p}+d_{p}}$ denotes the combined scale which for the protons is
associated with cyclotron resonance [7]. Here $d_{p,e}$ is the inertial length
where the particles decouple from the magnetic field and $\rho_{p,e}$ is the
gyration radius of the particles about the magnetic field direction. The trace
magnetic fluctuations show two power laws, while the compressive fluctuations
from three different measurement methods show a flattening between the ion
inertial and kinetic scales [53]. To estimate the compressive magnetic
fluctuations we use fluctuations in the magnitude of the magnetic field [e.g.
18]. The density fluctuations are shown from both FPI-DES and the spacecraft
potential method. The FPI measurement flattens at $f_{sc}>3$ Hz due to
instrumental noise while the spacecraft potential measurement can be used
throughout the entire range of interest before becoming noisy at 40Hz. The
green shaded area denotes the scales of interest of this study
[$f_{\text{min}},f_{\text{max}}$].The dark green area denotes frequencies
which are in the scales of interest but where the FGM or FPI-DES signals are
at the noise level. The noise level of the spacecraft potential is outside the
scales of interest and denoted by the grey shaded area. There are no
indications from the Fourier spectra that there are any spin tones or any
higher harmonics present although in the cross correlation spectra in Figure
2f there is a strong correlation near the second harmonic of the spin tone.
Therefore there may be some unphysical effect on the cross correlation however
these are very narrow in frequency and we expect the limitation to be smaller
as the frequency increases. Typically, magnetic field measurements on MMS have
spin tones at the first and second harmonics. The measured spacecraft
potential in the solar wind usually shows spin tones up to 1Hz due to the
sunlit surface changing as the spacecraft spins. However these have been
removed before calibrating to density. Some residual effect may exist at some
higher harmonics of the spin tone however this is unlikely to affect our
measurements as our frequency range of interest is above 2Hz which is at the
40th harmonic of the spin tone.

The ratio of the PSD of the magnetic field magnitude fluctuations to the trace
fluctuations is shown in Figure 2e. This gives a measure of the magnetic
compressibility which increases in the sub-ion range approaching a value of
1/3. A value of 1/3 indicates equal power in the compressible and the two
transverse components and is denoted by the orange line. The black curves
denote the estimation from Fourier analysis and the thick red lines denote the
estimation from wavelet analysis [54] which has been averaged in time. In
figures 2e-g a second y-axis is presented showing the spacecraft frame
frequencies obtained from the MSR analysis which is discussed later. These
points show two distinct populations of waves, and are presented here for
comparison with the spectral estimators.

Figure 2f shows the cross-correlation spectra;

$CC=\frac{\mathfrak{R}(\tilde{n}_{e}\tilde{B}^{*})}{|\tilde{n}_{e}||\tilde{B}|}$
(4)

of the density and the magnetic field magnitude where the tilde symbols denote
the Fourier transform (or the wavelet transform) and the asterisk denotes the
complex conjugate. Using this definition, -1 denotes anti-correlation and +1
denotes correlation. At inertial scales, the fluctuations are anti-correlated,
consistent with slow waves or pressure balanced structures [6], at smaller
scales the anti-correlation persists consistent with the polarization of the
kinetic Alfvén wave. There are also narrowband regions in frequency which
exhibit positive correlations. In the low-frequency range, these appear at the
second harmonic of the spacecraft spin (the spacecraft spin frequency and the
second harmonic are indicated by the vertical cyan lines) and are unlikely to
be physical. At larger frequencies there are some regions with positive
correlations, these cannot be resolved by the wavelet spectrum as the
frequency resolution is too low. A positive correlation is a characteristic of
the magnetosonic family of waves rather than the Alfvén wave family. Some
higher frequency points in blue may be associated with positive correlations
but it is difficult to make this distinction especially as the majority of the
high frequency points are in the region where the magnetic field is noisy
$f_{sc}>5$Hz. Figure 2g shows the ratio of normalized density to perpendicular
magnetic fluctuations [13, 55]. The kinetic Alfvén normalization is as
follows;

$\delta\mathbf{B}^{\prime}=\delta\mathbf{B}/B_{0},$ (5)

$\delta
n_{e}^{\prime}=\left(1+\frac{T_{i}}{T_{e}}\right)^{1/2}\frac{c_{s}}{v_{A}}\left[1+\left(\frac{c_{s}}{v_{A}}\right)^{2}\left(1+\frac{T_{i}}{T_{e}}\right)\right]^{1/2}\frac{\delta
n_{e}}{n_{0}},$ (6)

This allows for the transverse components of the magnetic field to be
estimated from

$\delta\mathbf{B}_{\perp}^{{}^{\prime 2}}=\delta\mathbf{B}^{{}^{\prime
2}}-\frac{\left(c_{s}^{2}/v_{A}^{2}\right)\left(1+T_{i}/T_{e}\right)}{1+\left(c_{s}^{2}/v_{A}^{2}\right)\left(1+T_{i}/T_{e}\right)}\delta
n_{e}^{\prime 2}.$ (7)

This relation comes from the nonlinear equations for kinetic Alfvén waves [56,
13]. Here we use the ion temperature measurement from OMNI and the other
parameters from MMS. For kinetic Alfvén waves the amplitudes of fluctuations
normalized in this manner are expected to be approximately equal $\delta
B_{\perp}^{\prime}\simeq\delta n_{e}^{\prime}$. For Fast/Ion Bernstein waves
this $\delta B_{\perp}^{\prime}\gg\delta n_{e}^{\prime}$ is expected. The mean
value of $\delta B_{\perp}^{\prime 2}\gg\delta n_{e}^{\prime 2}$in the scale
of interest (before instrumental noise becomes significant) is 0.13 consistent
with the results of [13], although there are regions where the individual
Fourier modes differ substantially from the mean. Magnetosonic fluctuations
would be expected to be smaller by more than one order of magnitude.

Figure 2h shows the magnetic helicity which is defined as;

$\sigma_{m}=\frac{2\mathfrak{I}\left(\tilde{B_{y}}\tilde{B_{z}}^{*}\right)}{|\tilde{B_{y}}|^{2}+|\tilde{B_{z}}|^{2}},$
(8)

where the $y$ and $z$ subscripts denote the GSE components. This method
assumes the wave-vector points into the $-x$ GSE direction. This assumption
will be justified later with our MSR analysis. The helicity is a measure of
the spatial rotation sense of the magnetic fluctuations about the wave-vector
direction. This can be used as a diagnostic for the type of plasma wave. The
helicity shows an increase before our region of interest which is commonly
interpreted as being due to kinetic Alfvén waves when the angle
$\theta_{BV}\sim 90^{\circ}$ [e.g. 57, 58, 59, 60]. However, at frequencies
larger than 5Hz, the helicity becomes zero. This is likely due to instrumental
noise. Low helicity could be is a characteristic of Ion Bernstein waves [61].
Alternatively, the contributions of oppositely directed KAWs can create a mean
helicity signal of zero if their power is balanced. We also note that
fluctuation levels in the slow solar wind are naturally weaker and may thus
create a weaker signal [62]. Nevertheless, our measurement of the helicity
suggests that fluctuations are KAW-like in the range $f_{sc}\in[0.5-5]$Hz. At
higher frequencies due to noise the measurement is inconclusive.

Figure 3 shows the wavelet spectra of the cross-correlation, the coherence,
and the local intermittency measure for both electron density and magnetic
field magnitude. The local intermittency measure (LIM) is defined as;

$LIM(t,\tau)=\frac{|\tilde{n}_{e}(t,\tau)|^{2}}{\langle|\tilde{n}_{e}(t,\tau)|^{2}\rangle_{t}}$
(9)

where the tilde denotes the wavelet transform, which is presented for both
$n_{e}$ and $|B|$ in figures 3 c and d respectively. In Fig 3a we see a
similar picture to the global measurements presented in Figure 2f with anti-
correlations dominating. However, there are small regions where positive
correlations are present. Rather than a different wave mode being present in
these regions, the wavelet coherence in 3b and the local intermittency measure
suggest that these are regions of low amplitude fluctuations where the
coherence is weak. The strong anti-correlation, the amplitudes and the
magnetic helicity suggest that KAW like fluctuations dominate in this range.
However, spectral estimators that use the magnetic field are limited to 5Hz
due to noise and we cannot infer conclusions from these estimators at higher
frequencies. Instead, we now use the electron density measurements which have
sufficient time resolution and sensitivity to study at frequencies far into
the sub-ion range $f_{sc}>$5Hz.

Figure 3: (a) wavelet cross correlation spectra between $|B|$ and $n_{e}$ (b)
wavelet coherence spectra between $|B|$ and $n_{e}$ (c) Local intermittency
measure for $n_{e}$ (d) local intermittency measure for $|B|$. The white line
denotes the cone of influence, where below this line the results are
unreliable due to edge effects.

_Results: MSR analysis_ Figure 4a,c shows the dispersion relation diagrams and
Fig 4b,d shows the angle the wave-vectors make with the mean magnetic field
direction calculated over the interval for the density measurements. For
clarity, these have been split into data points which are below the proton
cyclotron frequency $\Omega_{ci}$ and those above. The results show a mixture
of plasma waves both above and below the ion cyclotron frequency consistent
with previous studies using the magnetic field [51, 63, 64].

For the context of the larger scales, we plot data points reported in [5]. The
study of [5] used Cluster [65] magnetic field and spacecraft potential data
with inter-spacecraft distances of 200km and the plasma parameters are
comparable to the MMS interval. The wave-vectors from both intervals are shown
to make an almost perpendicular angle with the magnetic field, consistent with
other studies at larger scales [66, 67, 28, 68, 63, 64].

Figure 4: Dispersion relations and propagation angles obtained from the MSR
technique for plasma frame frequencies $<\Omega_{ci}$ (a,b) and $>\Omega_{ci}$
(c,d). The blue (orange) points denote the magnetic field (density) data
points from Cluster for solar wind with similar conditions. The black points
denote the data from our MMS interval. The errors are based on the uncertainty
in the wave-vector and the standard deviation of the ion velocity. The lines
show the solutions to the hot-plasma dispersion relations.

The Cluster data show that the fluctuations have low frequencies compared to
$\Omega_{ci}$ consistent with KAW turbulence, and the slightly larger spread
in the density points was interpreted as being due to a superposition of KAWs
and kinetic slow waves [5, 11]. We note that advected structures with no
intrinsic plasma-frame frequencies would exhibit a similar low-frequency
signature. Structures such as magnetic holes or pressure balanced structures
would also be able to explain the anti-correlations between density and
magnetic field magnitude. The lower frequency fluctuations from MMS presented
in Fig 4a show that there is an increase in the frequency at the smaller
scales consistent with the KAWs. This is plotted along with the dispersion
relation of KAWs obtained from the New Hampshire Dispersion relation Solver
[69] for the measured plasma conditions (where the OMNI ion $\beta$ is used)
and the measured mean propagation angle of $88.4^{\circ}$. The dispersion
relation for the KAW at an angle of $89.9^{\circ}$ is also plotted and shows
better agreement with the data. We emphasize that this plotted angle of
$89.9^{\circ}$ is within the standard deviation of the propagation angle.
Kinetic Alfvén waves are very sensitive to the propagation direction and an
uncertainty of a few degrees has a large impact on the solutions from linear
theory. The wave-vector of the fluctuations also make a small angle with the
ion bulk velocity direction of about $\sim 10^{\circ}$. Figure 5 shows the
unit vectors of the recovered wave-vectors at each frequency and the unit
vectors of the mean magnetic field direction, the mean bulk velocity, and the
inter-spacecraft baselines. This figure is presented in mean field aligned
coordinates with $\textbf{e}_{\parallel}=\mathbf{B}/B_{0}$, $\textbf{e}_{\perp
1}=\textbf{e}_{\parallel}\times\mathbf{V_{sw}}/|\mathbf{V_{sw}}|$. So that the
velocity is primarily in the negative $\textbf{e}_{\perp 2}$ direction. This
demonstrates that the wave-vectors associated with the most energetic
fluctuations are almost perpendicular to the mean magnetic field and are in
the direction of the mean bulk velocity. This observation justifies the
assumption used for measuring the magnetic helicity for this interval.

Figure 5: Three dimensional representation of the different directions
relevant for this study. The grey line denotes the mean magnetic field
direction in the $\mathbf{e}_{\parallel}$ direction. The thick black line
denotes the mean bulk velocity direction which is approximately perpendicular
to the magnetic field direction and is predominantly in the negative
$\mathbf{e}_{\perp 2}$ direction. The thick dashed lines denote the unit
vectors of different inter-spacecraft baselines. The thin cyan and magenta
lines denote wave vectors of the recovered IBWs and KAWs respectively which
make an approximately perpendicular angle with the mean magnetic field
direction and a small angles $\sim 10^{\circ}$ with the bulk flow direction.
For better presentation projections of the different unit vectors are shown on
the different planes.

For frequencies larger than the cyclotron frequency the fluctuations occur
predominantly near the third harmonic of the cyclotron frequency. This
suggests that some fluctuations follow the dispersion relation of IBWs [70].
In figure 6 the peak power of the fluctuations is obtained and is displayed as
a function of the wavenumber. Both the KAW like fluctuations and the IBW like
fluctuations have comparable power across all scales, however the IBWs have a
slightly steeper spectral index.

Figure 6: Peak power of the fluctuations at the recovered wave-vectors. Red
denote the IBWs while blue denote KAWs. The vertical error bars are obtained
from the $95\%$ confidence interval for 1024 degrees of freedom (256 windows,
4 spacecraft), the horizontal error bars are determined for a plane wave as
per [5]. Power law fits are performed to both fluctuation types and the error
is obtained from the residuals of the least squares fit.

Figure7, shows the properties of the linear kinetic Alfvén waves and several
harmonics of the ion Bernstein waves obtained from NHDS [69]. The dispersion
relations, linear damping rates, compressibility, magnetic compressibility,
and the phase angle between the magnetic field and density fluctuations is
shown. Here $\pm 180^{\circ}$ corresponds to anti-correlation, while
$0^{\circ}$ corresponds to a positive correlation. The large damping rate of
the IBW at the proton cyclotron frequency possibly explains why we do not
measure any points at this frequency in Fig4. Their linear damping rate is too
large to exist for more than a few wave periods. This could mean that the
damping rates are much stronger than any driving process so that they cannot
be generated, or that they do not exist as they are not excited at all.
However, higher-order ion-Bernstein branches are less severely damped and thus
able to exist. For these plasma conditions i.e. highly oblique wave-vectors,
and the plasma $\beta$ values the waves tend to avoid crossing
$\omega=\Omega_{ci}$ as the damping rates become large for these waves. This
is different from the solutions presented in [71] where the solutions cross
$\omega=\Omega_{ci}$ for smaller propagation angles
($\theta_{kB}<88^{\circ}$), for a larger plasma $\beta$ and a larger ion to
electron temperature ratio than is measured here. The compressibility of both
the KAWs and the IBWs with $k_{\perp}\gg k_{\parallel}$ increase at larger
wavenumbers which are consistent with the isotropy observed in the power of
the compressive and transverse components observed in Figure 2e, and in [12].
For KAWs the density and magnetic field fluctuations are anti-correlated where
the opposite is true for the ion Bernstein waves. However, in the sub ion
range the relative power of density fluctuations compared to the compressive
magnetic field is very large. Considering this observation linked with the
fact that the magnetic field instrument is nearly at the noise floor it is
difficult to identify positively correlated fluctuations through our spectral
estimators.

Figure 7: Linear Theory properties of the KAW (solid black line) and the
magnetosonic/Ion Bernstein branches, from top to bottom. The dispersion
relations (real part of the frequency), the damping rates, The ratio of
density to compressible magnetic amplitudes, the magnetic compressibility, the
phase angle between the density and compressible magnetic flcutuations.

The location of the wavenumber with the largest power poses an important
limitation for our method. So this method preferentially selects those
wavemodes with the largest power in the signal. In figure 8 we present a one-
dimensional cut of the power spectrum through $k_{\perp 2}$, which is
approximately the flow direction, at the smallest wavenumbers for the other
two components. We find these one-dimensional cuts for different spacecraft
frame frequencies and stack these results into the contour plot. We over-plot
are the relation expected for an advected structure (based on Taylor’s
hypothesis), for an Alfvén wave ($v_{sw}+v_{A}$) and a fast magnetosonic wave
($v_{sw}+v_{F}$ where $v_{F}=\sqrt{v_{A}^{2}+c_{s}^{2}}$). Although the
largest power is in fluctuations with low phase speeds i.e. KAW and advected
structures, there is a non-negligible component of power in between the curves
for the fast magnetosonic speed. This suggests that while the low frequency
components are dominant there are contributions from other types of
fluctuations. These may be Ion Bernstein modes as suggested in Fig 8, however
there could be contributions from sideband modes [e.g. 72, 73], where wave-
wave interactions have broadened the power distribution. These side-band
fluctuations may retain the properties of their parent waves e.g.
polarizations, fluctuation amplitudes however their frequencies are different
to their parent waves frequencies. Another possibility is that the broadening
is due to fluctuations in the velocity. In the Doppler shift equation large
fluctuations in the velocity can cause broadening in frequency [e.g. 74, 75].

Figure 8: Power distribution of fluctuations in the ($\perp 2$) flow direction
and spacecraft frame and the flow direction. Also indicated on the plot are
the bulk speed (red solid line) , the sum (and difference) with the Alfvén
speed (cyan dotted) and the magnetosonic speed (green dot dashed).

_Results:Intermittency_ Figure 9 shows the scale-dependent kurtosis which is
defined for increments of the measured density as;

$\kappa(\tau)=\frac{\langle\delta n_{e}^{4}\rangle}{\langle\delta
n_{e}^{2}\rangle^{2}}.$ (10)

where the increments are calculated from time lags $\delta
n_{e}(t,\tau)=n_{e}(t+\tau)-n_{e}(t)$ from a single spacecraft or a spatial
lag between two spacecraft $\delta
n_{e}(\lambda_{1,2})=n_{e}(\lambda_{1})-n_{e}(\lambda_{2})$. The single
spacecraft measurement gives a measure of the fluctuations along the bulk flow
direction, while different spacecraft pairs measure the fluctuations along
different baselines.

A kurtosis of 3 indicates non-intermittent Gaussian fluctuations and larger
values indicate that there are heavier tails in the probability distribution
function due to intermittent large amplitude fluctuations. We remove 4 seconds
at the edges of the time interval to avoid any large fluctuations associated
with the interpolation. We also remove outliers that are not sufficiently
sampled in the signal that can have an undue influence on the value of the
kurtosis due to the finite signal length. To do this we follow the scheme of
[76], removing the largest fluctuation until the value of the kurtosis
converges.

The time-lagged kurtosis is seen to increase, reaching a maximum near proton
scales but then decreases to a value $\sim 12$ in our range of interest and
does not show a strong dependence on the timescale. At higher frequencies,
instrumental noise and other effects such as dust strikes [77] affect the
measurement. Figure 9 also shows the kurtosis from the spatial lags indicated
by the symbols. These are converted to a time lag by using the measured ion
velocity. Although these values are in the noise region for the time lags,
they are likely to be more robust to the effects of noise than the time lags
[24] as noise is uncorrelated between spacecraft. Generally the difference in
the time and space lag could be due to a number of reasons. Its could
represent a breakdown in Taylor’s frozen in flow hypothesis due to the
presence of waves or structures that have evolution time scales faster than
the advection time over the spacecraft. This would be consistent with waves
with a high phase speed such as Ion Bernstein waves which are part of the
magnetosonic family of waves and could violate the hypothesis [e.g 78].
Alternatively, structures may merge or evolve on at faster timescales than the
advection time. One possible instrumental limitation is that the instruments
may have insufficient amplitude resolution to resolve the fluctuations between
spacecraft. The root mean squared values of the density fluctuations for the
different spacecraft are;
$[(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)]=[0.15,0.10,0.13,0.10,0.11,0.11]cm^{-3}$
while the RMS value of the time lag for the same scale from a time lag is
$0.03cm^{-3}$. This is similar to what was observed in the magnetic field by
[24]. Therefore, as the fluctuations measured from the spatial lags are larger
in terms of amplitude the measurement between the two spacecraft should be
less susceptible to noise. Another possibility is that the measured kurtosis
is direction dependent [79, 80, 81]. Lastly, a sampling effect related to the
bulk flow direction may lead to a situation in which we sample more structures
in this direction than along the spacecraft separation directions [82, 83,
84]. We notice that different baselines have different values of kurtosis with
the spacecraft pairs that are separated along the mean-field direction
(1,2),(1,3),(1,4) having lower values than (2,3),(2,4), and (3,4). The
geometry of this system is shown in fig 5. This is consistent with either IBWs
or KAWs randomizing the electron density fluctuations in the parallel
direction, while in the direction of wave propagation there is little or no
effect. The observations show that the kurtosis in the parallel direction is
reduced and the small error bars give us confidence that this is reduction is
physical. We note that a probe failure on MMS4 means that the spacecraft
potential measurement is made with two probes rather than the four probes on
the other spacecraft. Therefore there may be some effect that causes the pairs
with spacecraft four to have different results. We do not expect the result to
be affected here as the pair (2,3) (predominantly in the flow direction) has a
larger kurtosis than both (1,2) and (1,3) (parallel direction).

Figure 9: Scale-dependent kurtosis of the density fluctuations. The cyan
shaded area denotes $3\sigma$ calculated from one hundred bootstrap re-
samplings. The green area denotes the timescales of interest. The points
denote the kurtosis calculated from spatially lagged spacecraft pairs.
Similarly the error bars denote the $3\sigma$ error.

_Summary/Conclusions;_ we investigate compressive fluctuations in the sub-ion
range of solar wind turbulence. At these scales, the density fluctuations
exhibit strong wave-vector anisotropy, and plasma frame frequencies which are
consistent with KAW fluctuations, but there is also a population of high-
frequency fluctuations consistent with higher harmonics of the ion Bernstein
mode. The power in both fluctuation types is comparable, however, the spectral
index for IBWs is steeper than for KAWs. The measurements of the cross-
correlation, magnetic helicity and the relative normalized amplitudes of
density and magnetic fluctuations and the MSR analysis all support the
interpretation that KAWs are present in this interval. The combination of all
methods gives strong evidence that KAWs are an important component of solar
wind turbulence. However, as the magnetic field becomes noisy near 5Hz we
cannot confirm the presence of IBWs through the methods involving magnetic
fields in the majority of the frequency range studied here. The presence of
IBWs is only shown in our MSR analysis. This may be due to the instrumental
limitations of the magnetic field measurement at high frequencies, or that
wavelet analysis has insufficient frequency resolution to resolve the IBWs.
This might also be due to the inherent nature of IBWs at small scales. The
normalized density fluctuations are much larger than the compressive magnetic
fluctuations, meaning that a cross correlation may not be able to reveal such
fluctuations. Alternatively, this could be due to spatial aliasing [85], a
limitation of wave-based analysis methods leading to the inability to
differentiate between two wave-vectors where $\textbf{k}\cdot\textbf{r}$ that
differ by an integer multiple of $2\pi$. Spatial aliasing can manifest itself
in two ways; the first is that a wave near the edge of the analyzed k-space
can cause a spurious peak which is also in the region of interest. Both the
physical peak and the aliased peak would have the same power. We investigate
each peak individually and find that this effect does not seem to be present.
Practically we use the spacecraft configuration to define the region of
k-space in which such a peak cannot cause an alias. This region of k-space is
similar to the Brillouin zone in solid state physics [e.g. 86, 87, 88]. The
other possibility for spatial aliasing is the existence of a wave outside of
this zone that causes a peak inside; a possibility which we cannot completely
rule out. However in turbulence we expect that waves at higher frequencies
(and higher k) have less power and would not cause any significant aliasing.
This hypothesis is supported by our results in Fig 2. However we cannot
completely rule out some aliasing effects especially at higher wavenumbers.
One possibility is that the Ion Bernstein waves observed here are in fact KAWs
that have undergone many wave-wave interactions. Wave-wave interactions can
cause the frequency to broaden [e.g. 89, 73], which can mean that the
frequency identification alone is a limited method of diagnosing wavemodes in
a turbulent plasma. This would be consistent with the approximate equal power
observed between the two modes in Figure 6. In this scenario the IBW like
fluctuations may in fact be KAWs whose frequency have been broadened by wave-
wave interactions or by velocity fluctuations. This may explain why in the
below 5Hz why we do not see strong signatures of IBWs. The low signal to noise
ratio of the magnetic field measurement may result in a positively correlated
signature however due to the low amplitude of one component may have a weak
coherence (which is observed in Fig 3ab). The MSR analysis of the density time
series allows these fluctuations to be revealed. Indeed according the power in
k-space as a function of k in the flow direction and the spacecraft frame
frequency in Fig 8 there is significant power not only for the zero frequency
modes but also at a range which includes the fast magnetosonic speed. The fact
that there is significant power which is not accounted for when taking the
peak power should be kept in mind when interpreting dispersion plots such as
those in Fig 4 and in the literature [see a review of the results in 73].

Ion Bernstein waves may provide an additional channel for turbulent heating
[70]. If confirmed, this finding would have a major impact on the
applicability of Taylor’s hypothesis as they are strongly dispersive. The
waves are observed at scales where there is a reduction in the scale-dependent
kurtosis. The kurtosis is smaller in the direction along the mean magnetic
field direction when compared to the flow direction. This is consistent with
the interpretation that in the solar wind wave activity has a randomizing
effect on the fluctuations leading to a reduction in the observed kurtosis.
Instability driven waves can also destroy intermittency in this manner,
however the KAWs discussed here are more likely generated by the cascade. We
propose that at large scales Alfvén waves destroy the intermittency in the
transverse components, as the cascade proceeds and waves develop large
$k_{\perp}$ they become more compressive and in the sub-ion range begin to
also affect the compressive intermittency. An outstanding question is how
waves from the turbulent cascade can cause a reduction in intermittency. One
possibility is that stochastic heating [90] changes the distribution of the
fluctuation amplitudes, which is predicted to lead to a decrease of the scale-
dependent kurtosis at around the dissipation scale [30] . However this process
has been mostly studied for Alfvénic turbulence, and it is still unclear how
it affects compressive fluctuations. Furthermore, this effect cannot explain
the plateau seen in the scale dependent kurtosis between ion and electron
scales [30] seen in Fig 9 [and also in 91]. Coherent wave-wave interactions
can also generate coherent structures [e.g. 29] increasing the overall
intermittency. Should structures be generated from waves they are still
different entities, with very different Fourier signatures. Should there be no
randomization, waves would interact and interfere leading to only large scale
structures being present. It seems reasonable to assume that some degree of
randomization is required in the cascade. The interaction of waves that are
incoherent with respect to any coherent structure could then cause a reduction
in global statistics such as the kurtosis.

Typically in the sub-ion range intermittency destroying waves may be
considered to come from an instability or another process external to the
turbulent cascade itself. However, following the suggestion that the majority
of intervals of solar wind plasma are unstable [92] it is not clear whether we
can disentangle these two ideas. For example, a current sheet could form
naturally in the turbulence, become unstable and excite waves. In this case,
we cannot truly separate the turbulent cascade from local kinetic plasma
processes. Although in the observations of [92] it is not clear whether the
instabilities observed are dynamically relevant as their inverse growth rates
in the majority of unstable time intervals are large compared to the
turbulence’ timescales. There is no indication in this interval of an
instability being present, however the ion temperature measurements from MMS
are not sufficiently accurate to rule this out. It is unclear where the Ion
Bernstein waves originate from, one possibility is that they also cascade from
large to small scales but the powers are very small at large scales, and they
only become important far into the sub-ion range. The linear damping of both
KAWs and IBWs increases with $\beta$. Therefore, we expect that these waves
play a smaller role in the magnetosheath and thus allow for a larger overall
kurtosis as observed [24]. Curiously, this might affect plasmas such as the
magnetosheath with higher $\beta$ and where waves such as those observed here
cannot exist may still satisfy Taylor’s hypothesis while lower $\beta$ plasmas
might not. Our dispersion relation result is also consistent with the interval
of [64] where a low $\beta$ interval of magnetosheath was investigated.
Finally, we remark that this study is limited to a limited scale range. To
overcome this limitation we have compared with Cluster data to investigate
larger scales. Ideally, all of the relevant scales should be sampled
_simultaneously_ which can only be achieved from a measurement perspective by
surpassing four-point measurements such as the proposed Helioswarm concept
[93].

###### Acknowledgements.

The datasets analyzed for this study can be found in the MMS science data
archive https://lasp.colorado.edu/mms/sdc/public/. The calibrated spacecraft
potential data are available at
https://www.iwf.oeaw.ac.at/en/research/research-groups/space-plasma-
physics/sc-plasma-interaction/mmsaspoc-data-analysis/ Analysis of the
spacecraft potential data at IWF is supported by Austrian FFG projects
ASAP15/873685. D.V. is supported by the STFC Ernest Rutherford Fellowship
ST/P003826/1 and STFC Consolidated Grant ST/S000240/1. R.N. was supported by
Austrian FWF projects I2016-N20. Z. V. was supported by the Austrian FWF
projects P28764-N27.

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