# Turbulent Energy Conversion Associated with Kinetic Microinstabilities
in Earth’s Magnetosheath

Harry C. Lewis Department of Physics, Imperial College London, London, SW7
2BW, UK  Julia E. Stawarz Department of Mathematics, Physics and Electrical
Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK  Lorenzo
Matteini Department of Physics, Imperial College London, London, SW7 2BW, UK
Luca Franci Department of Mathematics, Physics and Electrical Engineering,
Northumbria University, Newcastle upon Tyne, NE1 8ST, UK  Kristopher G. Klein
Lunar and Planetary Laboratory and Department of Planetary Sciences,
University of Arizona, Tucson, Arizona 85721, USA  Robert T. Wicks Department
of Mathematics, Physics and Electrical Engineering, Northumbria University,
Newcastle upon Tyne, NE1 8ST, UK  Chadi S. Salem Space Sciences Laboratory,
University of California, Berkeley, California 94720, USA  Timothy S. Horbury
Joseph H. Wang Department of Physics, Imperial College London, London, SW7
2BW, UK

###### Abstract

Plasma in the terrestrial magnetosheath is characterised by very weak
particle–particle collisions, so kinetic microinstabilities are thought to be
responsible for regulating the thermodynamics of the plasma. By exciting
electromagnetic waves, these instabilities redistribute free energy in
velocity space, moulding the velocity distribution function (VDF) into a lower
energy state. In the high-beta magnetosheath, relatively small perturbations
to the VDF can easily excite instabilities compared to in the low-beta inner
heliosphere. Since magnetic fields cannot do work on the particles, electric
fields mediate energy exchange between the electromagnetic field and the bulk
fluid properties of the plasma. We investigate signatures of non-ideal energy
conversion associated with turbulent fluctuations in the context of electron
and ion temperature anisotropy–beta instabilities, utilising over 24 hours of
data spread over 163 distinct intervals of in situ magnetosheath observations
from Magnetospheric Multiscale (MMS). We find that average energy conversion
into fluid flow is enhanced along instability boundaries, suggesting that
turbulence is playing a role in how free energy is redistributed in the
plasma. The work enables a quantification of the energetics which are
associated with the role of kinetic microinstabilities in regulating
collisionless plasma thermodynamics. This work provides insight into the open
question of how specific plasma processes couple into the turbulent dynamics
and ultimately lead to energy dissipation and particle energisation in
collisionless plasmas.

To be added.

##  1 Introduction

Plasma turbulence is the predominant process by which fluctuation energy is
transferred from injection at large scales to dissipation at microphysical
scales. This phenomenon is known as the turbulent cascade and operates in a
wide range of astrophysical plasma environments (Matthaeus & Velli, 2011;
Galtier, 2018; Stepanova et al., 2022), such as (but not limited to) planetary
magnetospheres and ionospheres (Guio & Pécseli, 2021; Saur, 2021), stellar
coronae (Cranmer et al., 2015; Zank et al., 2021), stellar winds (Carbone,
2012; Bruno & Carbone, 2013; Verscharen et al., 2019; Adhikari et al., 2021),
the interstellar medium (Fraternale et al., 2022; Linsky et al., 2022), and
beyond (Subramanian et al., 2006; Evoli & Ferrara, 2011; Ruszkowski &
Pfrommer, 2023). Despite decades of intensive research, the nature of the
kinetic-scale interactions which mediate dissipation in collisionless space
plasmas are still poorly understood (e.g. Alexandrova et al., 2013; Kiyani et
al., 2015; Chen, 2016; Sahraoui et al., 2020; Matthaeus, 2021; Smith &
Vasquez, 2021; Schekochihin, 2022; Marino & Sorriso-Valvo, 2023).

The magnetosheath is a natural laboratory for collisionless plasma turbulence
(e.g. Alexandrova et al., 2008; Zimbardo et al., 2010; Matteini et al., 2016;
Chen & Boldyrev, 2017; Chhiber et al., 2018; Yordanova et al., 2020;
Rakhmanova et al., 2021), comprised of decelerated, heated, and compressed
solar wind plasma which has been deflected around Earth’s magnetosphere.
Magnetosheath dynamics are influenced by solar wind turbulence as it is
processed by the bow shock, as well as fluctuations driven by the shock itself
(Huang et al., 2017; Sahraoui et al., 2020; Rakhmanova et al., 2021). In
addition to strong fluctuations, the magnetosheath is host to complex velocity
distribution functions (VDFs) which deviate significantly from a Maxwellian
distribution Shuster et al. (2019). Like the turbulent fluctuations, the shape
of the VDF is influenced by the processing of the shock as well as
fluctuations in the magnetosheath. Nonthermal VDFs are susceptible to
instabilities, which scatter free energy in velocity space, driving the plasma
to relax into a state closer to thermodynamic equilibrium and exciting
electromagnetic wave modes (Gary, 1993; Treumann & Baumjohann, 1997).

In collisionless plasmas, internal energy needs to be redistributed via
wave–particle interactions, which requires that free energy is used to grow
the wave. There are various ways by which free energy can be scattered, but
the common factor is that, since magnetic fields ($\mathbf{B}$) cannot do work
on particles, energy transfer occurs via interactions with the electric field
($\mathbf{E}$). Energy exchange between the electromagnetic field and bulk
flow energy is mediated by $\mathbf{j}\cdot\mathbf{E}$, where
$\mathbf{j}=\sum_{s}q_{s}n_{s}\mathbf{v}_{s}$ is the total current density,
$q_{s}$ is the charge, $n_{s}$ is the number density, $\mathbf{v}_{s}$ is the
bulk flow velocity, and $s$ denotes the species. (Baumjohann & Treumann,
1996). Taking the contribution from the electron-frame electric field
$\mathbf{E}^{\prime}=\mathbf{E}+\mathbf{v}_{e}\times\mathbf{B}$ gives the
Joule dissipation, $\mathbf{j}\cdot\mathbf{E}^{\prime}$ (Zenitani et al.,
2011a; Vörös et al., 2019), which is the collisionless analogue of Ohmic
heating, $\eta j^{2}$. This term is important for collisionless dissipation
(Birn & Hesse, 2005, 2009; Zenitani et al., 2011b) because it encapsulates
energy conversion related to non-ideal electric fields, which are capable of
decoupling particle motion from $\mathbf{B}$. In contrast to $\eta j^{2}$,
which is positive by definition, $\mathbf{j}\cdot\mathbf{E}^{\prime}$ is
reversible and therefore sign-indefinite, although space plasma observations
typically find it to be positive on average (Matthaeus et al., 2020). Positive
values correspond to transfer from fields to flow (loading), whereas negative
values correspond to conversion from the flow to the fields (generating) (Birn
& Hesse, 2005). Within the magnetosheath, $\mathbf{E}^{\prime}$ is dominated
by $\mathbf{E}_{P_{e}}=-\frac{1}{n_{e}e}\nabla\cdot\mathbf{P}_{e}$ (the
pressure term of generalized Ohm’s law, where $\mathbf{P}_{e}$ is the electron
pressure tensor and $e$ is the elementary charge) at scales accessible to
spacecraft measurements (Vörös et al., 2019; Stawarz et al., 2021; Lewis et
al., 2023). A key question in the study of collisionless plasma turbulence and
its associated dissipation is understanding the origin of both the positive
and negative energy conversion signatures and how they may relate to specific
fundamental plasma processes.

Prior studies have demonstrated that temperature anisotropy
($T_{s,\perp}/T_{s,\parallel}\neq 1$) is constrained by the parallel plasma
beta $\beta_{s,\parallel}=2n_{s}k_{B}T_{s,\parallel}\mu_{0}/B^{2}$, where
$B=||\mathbf{B}||$ is the magnetic field strength, and $T_{s,\parallel}$ and
$T_{s,\perp}$ are the temperature parallel and perpendicular to the local
$\mathbf{B}$, respectively. Prior studies of the solar wind and magnetosheath
have observed this effect for the protons (Gary et al., 1995, 1997; Hellinger
et al., 2006; Maruca et al., 2012; Matteini et al., 2013; Maruca et al., 2018;
Bandyopadhyay et al., 2022a) and electrons (Gary et al., 2005; Xu & Chen,
2012; Zhang et al., 2018; Graham et al., 2021). The constraint put on
$T_{s,\perp}/T_{s,\parallel}$ by $\beta_{s,\parallel}$ is thought to be linked
to a number of different instabilities (Gary, 1993), for which functional
forms have been developed to parameterise the growth rate contours as a
function of dimensionless parameters $S_{e}$, $\alpha_{e}$ (electrons) and
$a$, $\beta_{0}$ (ions). These curves are expressed as
$T_{e,\perp}/T_{e,\parallel}=1+S_{e}\>\beta_{e,\parallel}^{-\alpha_{e}}$ (Gary
& Wang, 1996) and
$T_{i,\perp}/T_{i,\parallel}=1+a(\beta_{i,\parallel}-\beta_{0})^{-b}$
(Hellinger et al., 2006), for the electrons and ions, respectively. When
plotted on the anisotropy–beta plane, these curves serve as boundaries of the
accessible region of parameter space, indicating that rapidly-growing kinetic
microinstabilities have a controlling influence on the shape of VDFs. Electron
parameter space is bounded by the electron whistler and oblique electron
firehose instabilities, whereas the ions are influenced by the ion cyclotron,
ion mirror, parallel ion firehose and oblique ion firehose instabilities. The
instability growth rate thresholds are derived using linear Vlasov theory
under the assumption of perturbations to single bi-Maxwellian components for
each species growing in a uniform background. This is in apparent
contradiction with the nonlinear, nonuniform conditions characteristic of
turbulence, however the local nonlinear timescales are expected to be much
greater than linear growth rates (Bandyopadhyay et al., 2022b), although this
is not always true, especially in the solar wind (Klein et al., 2019a, 2021).
Bandyopadhyay et al. (2022b) found that in extreme regions of anisotropy–beta
parameter space this condition is reversed, providing a possible explanation
as to why linear theory is applicable to turbulent conditions such as those in
the magnetosheath.

The links between turbulence and small-scale instabilities have recently
started to be explored for the protons, for example in hybrid simulations of
Alfvénic turbulence (Markovskii et al., 2019; Bott et al., 2021) and in the
solar wind (Opie et al., 2023). However, this is still an open problem. The
resonant interactions responsible for particle energisation have been the
subject of detailed studies in recent years (e.g. Howes et al., 2017; Klein et
al., 2017; Chen et al., 2019; Verniero et al., 2021; Afshari et al., 2021;
Jiang et al., 2024), however they are difficult to investigate statistically
because of their purely kinetic nature and how complex it is to look at
individual features in the VDF. Instead, we adopt a fluid approach using
$\mathbf{j}\cdot\mathbf{E}^{\prime}$ to assess the net transfer of energy to
and from the particles, neglecting how the dissipation takes place in detail
on a particle-by-particle basis. In this letter, we look for the first time at
how this fluid heating step is organised with respect to the plasma
microinstabilities, to highlight how turbulent fluctuations are interacting
with the kinetic processes which regulate the plasma thermodynamics.

##  2 Data

Magnetospheric Multiscale (MMS) (Burch et al., 2016) is a multi-spacecraft
mission launched in 2015 to study magnetic reconnection in the geospace
environment (Fuselier et al., 2016; Tooley et al., 2016). The mission features
four probes orbiting in a tetrahedral formation, containing identical
instrument suites which measure electromagnetic fields (Torbert et al., 2016;
Russell et al., 2016; Lindqvist et al., 2016; Ergun et al., 2016) and VDFs
(Pollock et al., 2016) with unprecedented measurement cadence. In its highest-
resolution “burst”-mode, MMS samples the electromagnetic field at up to 8192
Hz and produces electron and ion VDFs at cadences of 0.03 s and 0.15 s
respectively. These rapid particle measurements facilitate in-depth
exploration of sub-ion scale turbulence that was not possible with prior
missions.

Our dataset consists of 24.4 hours of burst-mode data from 163 intervals of
continuous magnetosheath measurements. Intervals were selected which are free
from large-scale inhomogeneities and are all the equivalent of multiple
correlation lengths of the turbulent fluctuations in duration. The intervals
range from around 3—43 minutes in length, with a mean of 9 minutes. The
interval list is derived from that used by (Stawarz et al., 2022), with
additions up until the end of the 2019–2020 MMS dayside season. The interval
list is provided as a CSV file in the supplementary material. Intervals are
aggregated to create a single time series of variables on the electron and ion
measurement cadence, as appropriate. Moments of the VDFs are used to compute
$\mathbf{j}=en_{e}(\mathbf{v}_{i}-\mathbf{v}_{e})$ where quasineutrality is
used, $e$ is the elementary charge, and $\mathbf{v}_{i}$ is interpolated onto
the electron timescale, since small-scale fluctuations in $\mathbf{j}$ are
dominated by $\mathbf{v}_{e}$ (Stawarz et al., 2021). $\mathbf{E}^{\prime}$ is
calculated on the electron instrument cadence, which requires averaging
$\mathbf{E}$ and $\mathbf{B}$ onto the same timescale as $\mathbf{j}$.
Instantaneous local values of $\beta_{s,\parallel}$ and
$T_{s,\perp}/T_{s,\parallel}$ are used, where the ion values are interpolated
onto the electron cadence under the assumption that small-scale fluctuations
in these parameters can be neglected. Research in the present study is
conducted using measurements at the electron instrument frequency (33.33 Hz).

Figure 1: Probability density plotted on the anisotropy–beta plane for our MMS
dataset, with an example time series. Panel (a) represents the electron plane,
where solid and dashed lines correspond to $\gamma=10^{-2}\,\Omega_{c,e}$ and
$\gamma=10^{-1}\,\Omega_{c,e}$ growth rate thresholds, respectively. Panel (b)
shows the ion plane, with $\gamma=10^{-2}\>\Omega_{c,i}$ instability
thresholds. Panels (c)–(e) illustrate motion through parameter space during
2017-09-28/20:44:30—20:48:51. Panel (c) shows the turbulent magnetic field.
Panels (d) and (e) show the temperature anisotropy of the electrons and ions,
respectively. Shaded areas represent instantaneous
$\gamma=10^{-2}\,\Omega_{c}$ instability thresholds for each species.

Figure 1 panels (a) & (b) show how microinstabilities constrain accessible
regions of anistotropy–beta parameter space in our dataset. Electron and ion
instability thresholds are plotted using the parametric curves defined in Gary
& Nishimura (2003) and Maruca et al. (2018), respectively, wherein it is
assumed that VDFs are anisotropic Maxwellians and that there is no anisotropy
for one species when computing the growth curves of the other. Despite nonzero
background non-Maxwellianity and correlation between
$T_{e,\perp}/T_{e,\parallel}$ and $T_{i,\perp}/T_{i,\parallel}$ within our
dataset, the plots in Figure 1 demonstrate that the vast majority of
probability density lies within instability growth rate thresholds of
$\gamma=10^{-2}\,\Omega_{c}$, as seen in previous studies (Maruca et al.,
2018; Graham et al., 2021). Comparing Figures 1 (a) and (b), the ions are
typically able to deviate further from their respective instability growth
rate curves, and crucially the ion probability density appears more diffuse
across parameter space, irrespective of instability thresholds. In both
panels, the greatest occupancy of parameter space is concentrated around
$T_{\perp}/T_{\parallel}\sim 1$ and $\beta_{\parallel}\sim$
$10^{-1}$—$10^{1}$, indicating the typical conditions in the magnetosheath.
Panels (c) to (e) show that turbulent fluctuations drive changes to both
anisotropy and instability thresholds, indicating that a typical interval
moves significantly around parameter space as opposed to filling an isolated
region close to its average. To highlight this, an animation illustrating the
trajectory of the example time series through anisotropy–beta space is
provided in the supplementary material.

##  3 Results

We investigate how energy conversion to/from the electromagnetic fields is
organised with respect to kinetic microinstabilities by plotting
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ and
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ on the anisotropy–beta
plane. To account for any large-scale variability between intervals, we
normalise the time series of $\mathbf{j}\cdot\mathbf{E}^{\prime}$ by its mean-
removed root mean square (rms)
$=\sqrt{\langle(\mathbf{j}(t)\cdot\mathbf{E}^{\prime}(t)-\langle\mathbf{j}(t)\cdot\mathbf{E}^{\prime}(t)\rangle)^{2}\rangle}$
on an interval-by-interval basis. The data in these units represents an energy
conversion fluctuation where a value of 1 corresponds to the characteristic
fluctuation amplitude for the interval from which a given data point
originates. Similar qualitative results are obtained based on unnormalised
values of $\mathbf{j}\cdot\mathbf{E}^{\prime}$.

Figure 2: Average amplitude of energy conversion signals distributed on
anisotropy–beta plane. (a)
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ in
$T_{e,\perp}/T_{e,\parallel}$–$\beta_{e,\parallel}$ parameter space with
instability thresholds of $\gamma=10^{-2}\,\Omega_{c,e}$ (solid line) and
$\gamma=10^{-1}\,\Omega_{c,e}$ (dashed line). (b)
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ on the same plane. (c)
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ in
$T_{i,\perp}/T_{i,\parallel}$–$\beta_{i,\parallel}$ parameter space with
instability thresholds of $\gamma=10^{-2}\,\Omega_{c,i}$. (d)
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ on the same plane. In all
panels, bins with fewer than 5 points in total or originating from fewer than
2 intervals are excluded.

Figure 2(a) shows that there is enhanced
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ – corresponding to a
greater mean amplitude of energy conversion signals – in regions susceptible
to (oblique) electron firehose and electron whistler instabilities.
Qualitatively, high $\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ is
most strongly concentrated around the firehose threshold, extending to
$\gamma\leq 10^{-2}\,\Omega_{c,e}$. In comparison, the enhancement associated
with the whistler instability begins at $\gamma\geq 10^{-2}\,\Omega_{c,e}$.
Interestingly, there is an enhancement along the low–$\beta_{e,\parallel}$
edge of the distribution for $T_{e,\parallel}>T_{e,\perp}$. Despite being
partially obscured in the full set of intervals, closer investigation revealed
that this feature is robustly present for different subsets of intervals.
Given that this signature is not in the vicinity of the standard instability
thresholds, the significance of enhanced
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ in this region of
parameter space is less clear. It could be related to a possible instrumental
bias shifting the PDF in anisotropy–beta space, but to determine this would
require a separate systematic study. Figure 2(b) shows that mean signed energy
conversion is positive across most of parameter space, and that it is
particularly strong near the firehose and whistler instability thresholds. The
centre of the distribution is weakly net-positive, accounting for the majority
of overall energy conversion and suggesting that there is an average
dissipation of electromagnetic energy into particle energisation. At
low–$\beta_{e,\parallel}$, there are regions of negative average energy
conversion which partially overlap with enhancements in
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$.

There are interesting comparisons to draw with how energy conversion is
organised across ion parameter space. Figure 2(c) shows that, as for
electrons, $\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ is enhanced
along instability boundaries. However, the enhancement is most pronounced
beyond the instability thresholds, suggesting that a faster growth rate is
required to overcome the effects which are driving the ion VDFs into unstable
regions. An enhancement on the low–$\beta_{i,\parallel}$ edge is also visible,
albeit partially obscured as in the electron plot. This similarity may be
expected because the time steps that are at $\beta<1$ are mainly driven by
large magnetic fields, appearing as low–$\beta_{\parallel}$ points on both
species’ planes. Figure 2(d) shows that
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ in ion parameter space is
more sign-indefinite than the electrons beyond the instability thresholds,
with average net conversion from fields to flow only being maintained in the
stable centre of the distribution.

Figure 3:  Probability density function of
$\mathbf{j}\cdot\mathbf{E}^{\prime}$ in each region for (a) the electrons and
(b) the ions. Markers represent the centre of histogram bins. The dashed grey
curve indicates a normal distribution with the same mean and standard
deviation as the entire dataset.

To investigate the statistical trends identified in anisotropy–beta parameter
space, we manually select and group bins into regions of apparent enhanced
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ and investigate their
properties. Certain regions have unclear boundaries (particularly those in ion
parameter space), which introduces some variability, however the main features
we highlight are not impacted by this. Note that the enhancements associated
with the ion cyclotron and ion mirror instabilities have been treated as a
single region. Figure 3 shows the probability density function (PDF) in each
manually-defined region for bins in the range $[-3.5,3.5]$ rms units, within
which the vast majority (98.10%) of points lie. For context, 54.77% of the
dataset has $\mathbf{j}\cdot\mathbf{E}^{\prime}>0$ rms units, but only 10.10%
has $\mathbf{j}\cdot\mathbf{E}^{\prime}\geq 1$ rms units and 6.58% has
$\mathbf{j}\cdot\mathbf{E}^{\prime}\leq-1$ rms units. All regions, including
the remainders, have nonzero skewness and high kurtosis. Panel (a) shows there
is a slight enhancement in the negative tails of the electron PDF relative to
the remainder, but a more significant enhancement on the positive side,
leading to net positive $\mathbf{j}\cdot\mathbf{E}^{\prime}$. The
low–$\beta_{e,\parallel}$ region has diminished positive tail, but the
heaviest negative tail of all the regions. Bins with
$|\mathbf{j}\cdot\mathbf{E}^{\prime}|\gtrsim 2$ rms units do not have enough
counts to discern a reliable trend. Panel (b) demonstrates that ion
instability regions are nearly identical to the remainder in the negative
tail, but are enhanced on the positive side. The low–$\beta_{i,\parallel}$
region is reminiscent of its electron counterpart because they largely share
similar points; the negative tails are heavier than positive, and the peak
value is smaller. These trends will be assessed quantitatively in the next
section.

##  4 Discussion

Region | Data (%) | $\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ | $\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ | Variance | Skewness | Kurtosis
---|---|---|---|---|---|---
Electron firehose | 2.48 | 0.159 | 0.779 | 2.00 | 3.01 | 79.20
Electron whistler | 0.56 | 0.150 | 0.721 | 1.76 | 5.25 | 164.99
Electron low–$\beta_{e,\parallel}$ | 0.14 | -0.059 | 0.755 | 3.15 | 15.96 | 648.70
Electron remainder | 96.82 | 0.104 | 0.583 | 0.97 | 1.89 | 94.50
Ion firehose | 0.54 | 0.127 | 0.683 | 1.67 | 2.92 | 61.60
Ion cyclotron/mirror | 0.44 | 0.155 | 0.713 | 2.23 | 5.55 | 224.16
Ion low–$\beta_{i,\parallel}$ | 0.67 | -0.040 | 0.667 | 1.79 | 1.33 | 824.73
Ion remainder | 98.35 | 0.106 | 0.587 | 0.99 | 2.09 | 85.51
Table 1: Mean net and amplitude of $\mathbf{j}\cdot\mathbf{E}^{\prime}$,
variance, skewness, and kurtosis by region.

There are a number of processes involving energy conversion which may be
related to kinetic microinstabilities. Turbulent fluctuations driving the
plasma towards an instability threshold by distorting the distribution would
be expected to give a positive signature. Growth of wave modes as a result of
instabilities might be expected to give a negative signature. Redistribution
of free energy in velocity space due to wave growth may not show up in
$\mathbf{j}\cdot\mathbf{E}^{\prime}$. In our dataset, the presence of strong
fluctuations in $\mathbf{j}\cdot\mathbf{E}^{\prime}$ along the unstable
boundaries of the distribution is indicative of a link between turbulence and
the underlying instability processes. Largely positive
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ across parameter space
indicates that we predominantly observe the extraction of energy from
turbulent fluctuations into the fluid flow. However, around the instability
thresholds we see a mixture of net-positive and net-negative bins,
particularly for the ions, which suggests we are also sampling part of wave
growth due to instabilities.

Table 1 gives the average values of $\mathbf{j}\cdot\mathbf{E}^{\prime}$ and
$|\mathbf{j}\cdot\mathbf{E}^{\prime}|$ in each manually-defined region as
introduced in Section 3. The three electron regions represent 3.18% of the
total dataset, including 3.04% associated with the electron firehose and
whistler instabilities. $\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ in
the electron instability regions shows that the average energy conversion
direction is from the fields to the fluid flow, and that the average value is
increased compared to the remainder. This is consistent with electron VDFs
being driven into unstable configurations by fluid heating associated with
turbulent fluctuations. The low–$\beta_{e,\parallel}$ region has net energy
conversion from the fluid flow into the fields, which suggests we are
observing the scattering of free VDF energy into wave modes. The fraction of
data associated with ion regions is lower at 1.65%, with only 0.98% of
measurements associated with enhancements in
$\mathbf{j}\cdot\mathbf{E}^{\prime}$ at the ion cyclotron and firehose
instabilities. $\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ in the ion
regions is similar to electron regions, with net positive energy conversion
along instability thresholds and net negative conversion at small
$\beta_{i,\parallel}$.

Investigating the average amplitude of signals tells us about how much energy
conversion is occurring regardless of direction, without positive and negative
values cancelling each other out. The electron firehose region has the
strongest $\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ enhancement,
with a 33.6% increase compared to the remainder of the data. The electron
whistler region has the smallest increase at 23.6%, but this is still larger
than the enhancement in any of the ion regions. The low–$\beta_{e,\parallel}$
region has a 29.5% increase, showing that more energy conversion is occurring
on average in this region than in the remainder, albeit on average from flow
to fields. Compared to the remainder, the ion instability regions have
enhanced $\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$, by 16.4% (ion
firehose) and 21% (ion cyclotron), indicating that smaller energy conversion
signatures are present around ion instability thresholds than at equivalent
electron boundaries. This implies that energy conversion fluctuations to and
from the ion VDFs are on average weaker where kinetic microinstabilities are
growing or acting compared to the electron case, possibly as a result of the
ions being more nonthermal due to their lower thermal velocity. Similarly to
the electron equivalent, the low–$\beta_{i,\parallel}$ region has negative
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ and increased
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$.

For electrons, MMS may not sample particle distributions fast enough to
resolve all dynamics. This, and clear net positive
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$, suggests we are
preferentially observing the driving of electron VDFs into unstable regions.
In contrast, ion dynamics take place on longer timescales, potentially
indicating why we can observe fluctuations more clearly, explaining why
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ is enhanced but
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ is more sign-indefinite in
unstable regions. Wave propagation may bias our statistics against observing
negative $\mathbf{j}\cdot\mathbf{E}^{\prime}$ in the electrons, as
Svenningsson et al. (2022) found most electron whistler waves are observed
away from their source region. Other instabilities may be the same, although
the non-propagating electron firehose mode is expected to dominate compared to
the propagating mode (Cozzani et al., 2023). However, waves resulting directly
from ion instabilities have been observed in the solar wind (Gary et al.,
2016; McManus et al., 2024). Additionally, due to Taylor’s hypothesis (Taylor,
1938), which is frequently applied to the magnetosheath (and verified to be
true for a subset of our intervals in Stawarz et al. (2022)), MMS generally
observes spatial slices due to plasma advection rather than temporal slices
due to evolution. To account for these effects, a simulation study into
$\mathbf{j}\cdot\mathbf{E}^{\prime}$ and kinetic microinstabilities would be
valuable to track how the instability cycle is linked to turbulence as plasma
evolves through the magnetosheath.

In this study, we focus on $\mathbf{j}\cdot\mathbf{E}^{\prime}$ as the
representation of non-ideal energy conversion between the electromagnetic
fields and particle energy. However, this mechanism is not the only pathway
towards dissipation (Matthaeus et al., 2020; Pezzi et al., 2021). In the
statistical formulation of energy transfer channels, the pressure–strain
interaction, $-(\mathbf{P}_{s}\cdot\nabla)\cdot\mathbf{v}_{s}$, encapsulates
energy transfer between fluid flow and random thermal motion (Yang et al.,
2017a, b; Chasapis et al., 2018). Since the collisionless energy transfer
quantities are sign-indefinite yet are expected to lead to net dissipation on
average, it would be interesting to understand how
$\mathbf{j}_{s}\cdot\mathbf{E}$ and
$-(\mathbf{P}_{s}\cdot\nabla)\cdot\mathbf{v}_{s}$ balance locally. However,
$-(\mathbf{P}_{s}\cdot\nabla)\cdot\mathbf{v}_{s}$ is a difficult quantity to
compute using observational data because the electron-scale pressure gradients
are frequently smaller than can be measured by particle instruments (Dahani et
al., 2024). Other examples of processes which can modify the internal energy
are the divergence of the vector heat flux density (Du et al., 2020), and
dissipation associated with an increase in entropy due to collisions (Liang et
al., 2019, 2020; Argall et al., 2022; Barbhuiya et al., 2024). Even in weakly
collisional plasmas, non-Maxwellian features in the VDF are subject to
intraspecies collisions as a result of the dissipation of strong velocity-
space gradients (e.g. Schekochihin et al., 2009; Pezzi et al., 2021). Such
entropy studies may provide a novel way of assessing the scattering of
particles in the distribution function associated with the instabilities.
Further studies may utilise future spacecraft missions or simulations to
expand on the results we have presented here and investigate the link between
energy conversion measures in the context of kinetic microinstabilities.

##  5 Conclusion

In this letter, we present a statistical investigation into non-ideal energy
conversion associated with kinetic microinstabilities, revealing enhanced
levels of $\mathbf{j}\cdot\mathbf{E}^{\prime}$ close to temperature
anisotropy–beta instability thresholds. In these regions, compared to the
remainder of the distribution, energy conversion has increased amplitude and
larger positive net value, indicating that we more often observe driving of
VDFs into unstable configurations than scattering of free energy associated
with the growth of wave modes. We find that the greatest enhancement in
$\langle\mathbf{j}\cdot\mathbf{E}^{\prime}\rangle$ and
$\langle|\mathbf{j}\cdot\mathbf{E}^{\prime}|\rangle$ occurs around the
electron firehose instability boundary. In contrast, the largest increase in
ion parameter space occurs beyond the ion cyclotron instability threshold. We
also highlight an additional region of interest, present in both parameter
spaces, along the low–$\beta_{\parallel}$ edge, at
$T_{e,\perp}/T_{e,\parallel}<1$ for the electrons and
$T_{i,\perp}/T_{i,\parallel}\sim 1$ for the ions. For both species, the
low–$\beta_{\parallel}$ edge has net negative energy conversion (with a
smaller absolute value than the net of the remainder), highlighting these
regions as areas where significant energy is being imparted into
electromagnetic activity. The origin of this effect is not yet understood.
Despite accounting for a small percentage of the overall observations, we have
shown that non-ideal energy conversion is enhanced when VDFs are configured
such that they are susceptible to anisotropy-driven kinetic
microinstabilities. We suggest that, while these processes are not responsible
for the large-scale energy budget of the magnetosheath, the instability
growth—excitation—scattering cycle is playing an important active role in how
turbulent fluctuations influence the plasma thermodynamics.

There are some unresolved questions which warrant exploration in a further
study. The relationship between kinetic microinstabilities and energy
conversion parallel and perpendicular to the local $\mathbf{B}$ is
interesting, because of the consequences for particle acceleration along field
lines. In addition, there are extra sources of free VDF energy arising due to
deviations from bi-Maxwellians which are unaccounted for by the
anisotropy–beta instability thresholds (Klein et al., 2018, 2019b). Some work
has been done to uncover how this feature is organised with respect to the
anisotropy-driven kinetic microinstabilities (Graham et al., 2021; Walters et
al., 2023), and future investigations should aim to link trends in the non-
Maxwellianity to energy conversion. Finally, other energy conversion measures
– such as the pressure–strain interaction, and the contribution to fluid
heating from different elements of $\mathbf{j}$ and various components of
$\mathbf{E}$ – may be treated with a similar analysis as in the present paper
in order to uncover their relationship to the regulation of plasma
thermodynamics.

H.C.L. and J.E.S. are supported by the Royal Society University Research
Fellowship No. URF\R1\201286. L.F. was supported by the Royal Society
University Research Fellowship No. URF\R1\231710 and Science and Technology
Facilities council (STFC) grant ST/W001071/1. H.C.L. and L.F. were supported
by the International Exchange Grant No. IES\R1\231251. K.G.K. was supported by
NASA Grant 80NSSC19K0912. J.H.W. was supported by STFC studentship
ST/X508433/1. This research was supported by the International Space Science
Institute (ISSI) in Bern, through ISSI International Team project #556 (Cross-
scale energy transfer in space plasmas). J.E.S. acknowledges useful
discussions as part of the ISSI International Team project #23-588 (Unveiling
Energy Conversion and Dissipation in Non-Equilibrium Space Plasmas). The
authors wish to thank the entire MMS Team for their work on the mission. The
data used in this study are publicly available through the MMS Science Data
Center at https://lasp.colorado.edu/mms/sdc/public/. Elements of data analysis
were performed using the pyrfu analysis package available at
https://github.com/louis-richard/irfu-python. H.C.L. thanks colleagues at
Imperial College London for valuable discussions.

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Variable substitutions can be specified using the \--option or -o command line
switches, e.g.

⬇

jobsub.py \--option beamenergy=5.3 -c alignment.conf 1234

This switch can be specified several times for multiple options or can parse a
comma-separated list of options. This switch overrides any configuration file
options.

##### Table (comma-separated text file)

Tables in the form of CSV files can be used to replace placeholders with the
-csv option. For the correct format, the following tools can be used:

  * •

export from LibreOffice with default settings (UTF-8, comma-separated, text-
field delimiter: ")

  * •

emacs org-mode table (see http://orgmode.org/manual/Tables.html)

  * •

use Atom’s _tablr_ extension

  * •

or the CSV file can be edited in a text editor of choice.

The following rules apply:

  * •

commented lines (starting with #) are ignored

  * •

first row (after comments) has to provide column headers which identify the
variables in the steering template to replace (case-insensitive)

  * •

requires one column labeled “RunNumber”

  * •

only considers placeholders left in the steering template after processing
command-line arguments and configuration file options

Strings can be passed by the user of double-quotes "" which also avoid the
separation by commas. A double-quote can be used as part of a string when
using the escape character backslash \\}infrontofthedouble-
quote.\parItisalsopossibletospecifymultipledifferentsettingsforthesamerunnumberbymakinguseofthefollowingsyntaxtospecifyasetorrangeofparameters.Curlybracketsindouble-
quotes\parameter{"{}"}canbeusedtoindicateaset(indicatedbyacomma\parameter{,})orrange(indicatedbyadash\parameter{-})ofparameterswhichwillbesplitupandprocessedoneaftertheother.Ifasetorarangeisdetected,theparameterplusitsvalueareattachedtothenameoftheconfigurationfile.Rangescanonlybeusedforintegervalues(withoutunits).However,asetorrangecanolybeusedforoneparameter,i.e.~multi-
dimensionalparametersscansarenotsupportedandhavetobe%****␣jobsub.tex␣Line␣200␣****separatedintoindividualCSVfiles.\par\begin{itemize}\tightlist\item\parameter{"{10,12-14}"}translatesto\parameter{10},\parameter{12},\parameter{13},\parameter{14}inconsecutivejobsforthesamerunnumber\item\parameter{"{10ns,20ns}"}tranlatesto\parameter{10ns},\parameter{20ns}inconsecutivejobsforthesamerunnumber\item\parameter{"string,with,comma"}translatesto\parameter{string,with,comma!}inonejob\item\parameter{"{string,with,comma}"!}whichtranslatesto\parameter{string},\parameter{with},\parameter{comma}inconsecutivejobsforthesamerunnumber\item\parameter{"\"stringinquotes\""}translatesto\parameter{"stringinquotes"}\end{itemize}\parIfarangeorsetofparametersisdetector,thenamingschemeoftheauto-
generatedconfigurationfilesisextendedfrom\parameter{MyAnalysis_run@RunNUmber@.conf}to\parameter{MyAnalysis_run@RunNUmber@_OtherParameter@OtherParameter@.conf}\parItmustbeinsuredbytheuserthattheoutputROOTfileisnotsimplycalled\parameter{histograms_@RunNumber@.root}butrather\parameter{histograms_@RunNumber@_OtherParameter@OtherParameter@.root}topreventoverwritingtheoutputfile.\par\hypertarget{example}{%\subparagraph{Example}\label{example}}\parTheCSVfilecouldhavethefollowingform:\par\begin{minted}[frame=single,framesep=3pt,breaklines=true,tabsize=2,linenos]{ini}#AnalysisExample.csv#Thisisanexample.RunNumber,ExampleParameter,AnotherParameter100,"{3-5}",10ns101,3,"{10ns,20ns}"\end{minted}\parUsingthistable,theplaceholders\parameter{@RunNUmber@},\parameter{@ExampleParameter@},and\parameter{@AnotherParameter@}inthetemplatefile\parameter{AnalysisExample.conf}wouldbereplacedbythevaluescorrespondingtothecurrentrunnumberandthefollowingconfigurationfileswouldbegenerated:\par\begin{itemize}\tightlist\itemAnalysisExample\\_run100\\_exampleparameter3.conf\itemAnalysisExample\\_run100\\_exampleparameter4.conf\itemAnalysisExample\\_run100\\_exampleparameter5.conf\itemAnalysisExample\\_run101\\_anotherparameter10ns.conf\itemAnalysisExample\\_run101\\_anotherparameter20ns.conf\end{itemize}\par\hypertarget{example-
usage-with-a-batch-
file}{%\subsubsection{ExampleUsagewithaBatch%****␣jobsub.tex␣Line␣250␣****File:}\label{example-
usage-with-a-batch-
file}}\parExamplecommandlineusage:\par\begin{minted}[frame=single,framesep=3pt,breaklines=true,tabsize=2,linenos]{ini}./jobsub.py-c/path/to/example.conf-
vDEBUG--
batch/path/to/example.sub\‚--subdir<run_number>\end{minted}\parAnexamplebatchfileisprovidedintherepositoryas\parameter{htcondor.sub}.Complicatedanderror-
prone\parameter{transfer_output_files}commandscanbeavoided.Itismuchsimplertosetanabsolutepathlike\par\begin{minted}[frame=single,framesep=3pt,breaklines=true,tabsize=2,linenos]{ini}output_directory="/eos/user/y/yourname/whateveryouwant/run@RunNumber@"\end{minted}\pardirectlyintheCorryvreckanconfigurationfile.

##  Appendix A Acknowledgements

Corryvreckan has been developed and is maintained by:

  * •

Morag Williams, University of Glasgow/CERN

  * •

Jens Kroeger, University of Heidelberg/CERN

  * •

Daniel Hynds, NIKHEF

  * •

Simon Spannagel, CERN

The following authors, in alphabetical order, have contributed to
Corryvreckan:

  * •

Matthew Daniel Buckland, University of Liverpool

  * •

Estel Perez Codina, CERN

  * •

Dominik Dannheim, CERN

  * •

Katharina Dort, University of Gießen/CERN

  * •

Adrian Fiergolski, CERN

  * •

Lennart Huth, DESY

  * •

Andreas Matthias Nürnberg, KIT

  * •

Florian Pitters, HEPHY Vienna

  * •

Paul Schütze, DESY

  * •

Tomas Vanat, CERN

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