# Local energy transfer rate and kinetic processes: the fate of turbulent
energy in two-dimensional Hybrid Vlasov-Maxwell numerical simulations

Luca Sorriso-Valvo1 Email address for correspondence<EMAIL_ADDRESS>Denise Perrone2,3 Oreste Pezzi4 Francesco Valentini4 Sergio Servidio4 Ioannis
Zouganelis2 & Pierluigi Veltri4 1Nanotec/CNR, Sede di Cosenza, Ponte P. Bucci,
Cubo 31C, 87036 Rende, Italy.
2European Space Agency, ESAC, Madrid, Spain.
3Department of Physics, Imperial College London, London SW7 2AZ, United
Kingdom.
4Dipartimento di Fisica, Università della Calabria, Ponte P. Bucci, Cubo 31C,
87036 Rende, Italy.

(?; revised ?; accepted ?. - To be entered by editorial office)

###### Abstract

The nature of the cross-scale connections between the inertial range turbulent
energy cascade and the small-scale kinetic processes in collisionless plasmas
is explored through the analysis of two-dimensional Hybrid Vlasov-Maxwell
numerical simulation (HVM), with $\alpha$ particles, and through a proxy of
the turbulent energy transfer rate, namely the Local Energy Transfer rate
(LET). Correlations between pairs of variables, including those related to
kinetic processes and to deviation from Maxwellian distributions, are first
evidenced. Then, the general properties and the statistical scaling laws of
the LET are described, confirming its reliability for the description of the
turbulent cascade and revealing its textured topology. Finally, the connection
between such proxy and the diagnostic variables is explored using conditional
averaging, showing that several quantities are enhanced in the presence of
large positive energy flux, and reduced near sites of negative flux. These
observations can help determining which processes are involved in the
dissipation of energy at small scales, as for example ion-cyclotron or mirror
instabilities typically associated with perpendicular anisotropy of
temperature.

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##  1 Introduction

The dynamical behavior of plasmas depends markedly on frequencies and scales.
At low frequencies and large scales, ions and electrons are locked together
and behave like an electrically conducting fluid; this is the regime of
magnetohydrodynamics (MHD). At higher frequencies and smaller scales,
electrons and ions can move relative to each other, behaving like two separate
and interpenetrating fluids: this is the two-fluid regime. However, if the
dynamics is supported by non-thermal effects and anisotropies in the velocity
space, the more complete kinetic theory shall be adopted. A notable example is
represented by the solar wind, a continuous, highly variable, weakly
collisional plasma outflowing at high speed from the Sun. This complex medium
is highly structured, with turbulent fluctuations on a wide range of scales,
from fractions of a second to several hours, thus covering all the above
dynamical ranges.

Spacecraft measurements have revealed that the solar-wind plasma is usually in
a state of fully developed turbulence (Marsch, 2006; Bruno & Carbone, 2013;
Alexandrova et al., 2013). As the turbulent fluctuations expand away from the
Sun, these decay and transfer energy to smaller scales. At scales comparable
with the ion gyro-radius or the ion inertial length, the MHD approximation
fails and kinetic processes involving field-particle interactions take place,
along with important cross-scale coupling (Schwartz et al., 2011).
Furthermore, since the plasma collisionality is quite low, the ion and
electron velocity distribution functions can exhibit significant non-
Maxwellian signatures. Kinetic processes manifest through the excitation of
plasma waves, temperature anisotropy, plasma heating, particle energization,
entropy cascade, excitation of plasma waves and so on (Vaivads et al., 2016).
Recently, a phenomenological description in terms of cascade in the velocity
space, due to highly structured particle distributions, has been proposed
(Schekochihin et al., 2016; Servidio et al., 2017). Observations based on the
high-resolution ion velocity distribution measurements by the Magnetospheric
Multiscale (MMS) mission in the Earth’s turbulent magnetosheath (Servidio et
al., 2017) have recently confirmed such description, opening a new pathways to
the understanding of plasma turbulence and dissipation in collisionless
plasmas. This cascade may efficiently transfer the energy towards smaller
velocity scales, thus producing finer structures in the particle velocity
distribution function. The presence of such structures may ultimately enable
and enhance the effect of collisions in dissipating the energy (Pezzi et al.,
2014, 2015a, 2015b, 2016a, 2016b; Pezzi, 2017). Numerical simulations
represent an indispensable tool for the understanding of plasma nonlinear
dynamics. The use of kinetic numerical simulations is crucial, indeed, when
kinetic processes occur in turbulent collisionless plasmas (Parashar et al.,
2009; Camporeale & Burgess, 2011; Servidio et al., 2015; Franci et al., 2015;
Cerri et al., 2016, 2017; Groselj et al., 2017; Franci et al., 2018). The
Eulerian hybrid Vlasov-Maxwell HVM code (Valentini et al., 2007; Perrone et
al., 2011) can be used to study the dynamics of a turbulent plasma with
typical solar wind conditions, in order to describe the cross-scale coupling
between the MHD turbulence and the small-scale kinetic processes (Valentini et
al., 2008; Valentini & Veltri, 2009; Valentini et al., 2010, 2011a, 2011b;
Perrone et al., 2011; Servidio et al., 2012; Greco et al., 2012; Perrone et
al., 2013; Servidio et al., 2014; Perrone et al., 2014a, b; Valentini et al.,
2014; Servidio et al., 2015; Valentini et al., 2016; Pezzi et al., 2017a, b).

In turbulence, at small scales, the energy level of the fluctuations becomes
too low to be investigated with poorly-resolved Lagrangian algorithms with
very low number of particles-per-cell (for a convergence study, see Haggerty
et al. (2017)). Eulerian-based numerical algorithms might be more suitable for
this scope. HVM simulations show that kinetic effects produce, close to
reconnection regions, a distortion of the ion velocity distribution, through
the generation of accelerated field-aligned beams, temperature anisotropy and
trapped particle populations. Moreover, departures from the Maxwellian shape
are more pronounced for heavy ions than for protons (Perrone et al., 2013,
2014b; Valentini et al., 2016), pointing out that the role of secondary ions
must be considered for understanding energy partition in plasmas (Marsch,
2006; Kasper et al., 2008; Araneda et al., 2009; Maneva et al., 2015).
Although the physical mechanisms of solar wind turbulence have been matter of
investigation for many decades, both using ‘in situ’ measurements and
numerical experiments, important problems concerning the interconnection
between the inertial and kinetic range processes, as well as the interplay
between physical space fluctuations and velocity space perturbations, are
still poorly understood.

Solar wind turbulence is characterized by intermittency (Burlaga, 1991;
Frisch, 1995), i.e. the spatial inhomogeneity of the energy transfer across
scales, that results in the generation of highly energetic, bursty small-scale
fluctuations with scale-dependent, non-Gaussian statistics (Sorriso-Valvo et
al., 1999). Such fluctuations are often referred to as coherent structures,
with phase synchronization among a certain number of scales (Greco & Perri,
2014; Lion et al., 2016; Roberts et al., 2016; Perrone et al., 2016, 2017). At
smaller scales, below the MHD range, a more variable level of intermittency is
observed, whose nature is still under debate (Alexandrova et al., 2008; Kiyani
et al., 2009). During the last years, the study of the intermittency and the
consequent research on small-scale structures have been performed by using
different methods. For example, wavelet analysis (Farge, 1992; Bruno et al.,
1999) provides the local estimation of the scale-dependent energy associated
with the magnetic fluctuations, thus allowing the identification of the highly
energetic intermittent structures. Alternatively, the Partial Variance of
Increments (PVI) technique (Greco et al., 2008) was used to identify
discontinuities, current sheets, and other intermittent features.

Recently, there is a compelling evidence that the kinetic processes are
enhanced in the proximity of the turbulence-generated structures, which carry
a larger amount of energy than the turbulent background. Measurements have
shown that both ions and electrons are energized in the proximity of the most
intense small-scale current sheets (Osman et al., 2011, 2012; Tessein et al.,
2013; Chasapis et al., 2015, 2017). Note that the same behavior has been
observed in HVM numerical simulations (Servidio et al., 2012; Greco et al.,
2012; Perrone et al., 2014b). More recently, even reduced models such as giro-
kinetics converged toward this general view (Howes, 2016; Howes et al., 2018).
The processes responsible for such different forms of energization may involve
magnetic reconnection (Retinò et al., 2007; Burch et al., 2016b), wave-
particle resonances, plasma instabilities (Matteini et al., 2013; Servidio et
al., 2014; Breuillard et al., 2016) and enhancement of collisions (Pezzi et
al., 2016a). These topics are currently very debated in the space plasma
community (Chen, 2016).

In order to understand the interconnections between the turbulent structures
and the small-scale kinetic processes, a more detailed description of the
local energy flux across scales should be defined for plasma turbulence. This
possible tool should take into account the complex nature of the nonlinear
coupling between velocity and magnetic field. Recent studies included the
contributions from cross-terms, typically associated to Alfvénic fluctuations,
to better identify regions of small-scale accumulation of energy (Sorriso-
Valvo et al., 2015).

In MHD, the energy transfer across scales obeys the Politano-Pouquet law (PP)
(Politano & Pouquet, 1998), which prescribes the linear scaling of the mixed
third-order moment of the fields increments. The proportionality factor is the
mean energy transfer rate $\langle\epsilon^{\pm}\rangle$, the brackets
indicating ensemble average. For a two-dimensional field, the standard version
of the PP law for the the mixed third-order moments $Y^{\pm}(l)$ is

$Y^{\pm}(l)=\langle|\Delta\mathbf{z^{\pm}}_{l}(\mathbf{r})|^{2}\,\Delta
z^{\mp}_{i,l}(\mathbf{r})\rangle=-2\langle\epsilon^{\pm}\rangle l\;.$ (1)

$\Delta\psi_{l}=\psi(\mathbf{r}+l\mathbf{r}/r)-\psi(\mathbf{r})$ is the
longitudinal increment of a generic field $\psi$ (e.g. the magnetic field
$\mathbf{B}(\mathbf{r})$) across the scale $l$; the subscript $i$ indicates
the generic component of the field, the field increment being estimated along
the same direction; $\mathbf{z^{\pm}}=\mathbf{v}\pm\mathbf{B}/\sqrt{4\pi\rho}$
are the Elsasser variables that couple the plasma velocity $\mathbf{v}$ and
the magnetic field $\mathbf{B}$, the latter being expressed in velocity units
through the plasma density $\rho$. The validity of the PP law has been first
verified in two-dimensional MHD numerical simulations (Sorriso-Valvo et al.,
2002), and later also in the solar wind (MacBride et al., 2005; Sorriso-Valvo
et al., 2007; Marino et al., 2008; MacBride et al., 2008; Marino et al.,
2012).

Based on the PP law (Equation 1), an heuristic proxy of the ‘local’ energy
transfer rate (LET) at the scale $l$ can be defined by introducing the
quantity:

$\epsilon_{l}(\mathbf{r})=-\frac{1}{2l}\left[|\Delta\mathbf{z^{+}}_{l}(\mathbf{r})|^{2}\,\Delta
z^{-}_{i,l}(\mathbf{r})+|\Delta\mathbf{z^{-}}_{l}(\mathbf{r})|^{2}\,\Delta
z^{+}_{i,l}(\mathbf{r})\right]\,.$ (2)

At a given scale, each field increment in the time series can thus be
associated with the local value of $\epsilon_{l}(\mathbf{r})$ (Marsch & Tu,
1997; Sorriso-Valvo et al., 2015, 2018). Note that the LET can be expressed in
terms of velocity and magnetic field as $\epsilon_{l}(\mathbf{r})\propto\Delta
v_{i}(\Delta v^{2}+\Delta b^{2})-2\Delta
b_{i}(\Delta\mathbf{v}\cdot\Delta\mathbf{b})$. The first right-hand-side term
is associated with the energy advected by the velocity fluctuations, and the
second to the velocity-magnetic field correlations coupled to the longitudinal
magnetic field fluctuations (Andrés & Sahraoui, 2017; Sorriso-Valvo et al.,
2018). Recently, by using Helios 2 data, it has been shown that the LET
correctly identifies the regions of space in the turbulent solar wind that
carry energy towards smaller scales. These regions are characterized by
enhanced kinetic processes, resulting in higher proton temperature (Sorriso-
Valvo et al., 2018). It is worth noting that in recent works, the third order
law in Eq. (1) has been extended to include characteristic plasma length
scales and compressibility (Andrés et al., 2018).

The correlation between the local temperature increase and the large amplitude
bursts of energy transfer suggests that the proxy defined in Eq. (2) might
play an important role in the understanding of solar wind turbulent
dissipation and heating. Moreover, the comparison between the properties of
the PVI and LET has shown that both methods are suitable for the description
of the turbulent energy cascade. Since the LET is a signed variable, in
principle it should also carry information about the possible direction of the
cross-scale energy flow, identifying possible local direct and inverse cascade
processes. The latter cannot be studied by using PVI or wavelet-based
techniques since these are positive-defined.

The aim of this paper is to explore the link between the MHD turbulent energy
cascade and the kinetic processes associated with deviations of the ion
velocity distributions from the Maxwellian equilibrium. To this aim, we make
use of the LET for interpreting high-resolution data from numerical
simulations of the Vlasov-Maxwell multi-component system. The present work is
organized as follows. In Section 2 the governing equations, the numerical
algorithm and the parameters of the Vlasov simulations will be presented. A
brief overview on the numerical results will be given. The use of the proxy of
the local energy transfer rate will be the topic of Section 3, while in
Section 4 we will discuss our conclusions.

##  2 HVM multi-component simulations

We use high resolution Vlasov simulations in order to simulate the turbulent
multi-species plasma of the solar wind. First, high spatial resolution is
needed in order to follow the evolution of the turbulent cascade up to
wavelengths shorter than ion characteristic scales, where kinetic effects
presumably govern the physics of the system. Second, high numerical resolution
in the velocity domain is required to investigate the distortions of the ion
distribution functions. The above two requirements are the main ingredients of
the highly-performing, extensively tested, HVM code
(http://fis.unical.it/hvm/).

###  2.1 Numerical method

We employ the HVM code which directly integrates the Vlasov equations for both
proton and $\alpha$ particle distribution functions while electrons are
considered as a massless isothermal fluid. The Vlasov equation for both ion
species is combined with the Maxwell system, by assuming quasi-neutrality and
by neglecting the displacement current. In particular Ampére and Faraday
equations and a generalized Ohm’s law are solved. HVM equations, written in
dimensionless form, are the following:

$\displaystyle\frac{\partial f_{s}}{\partial t}+{\bf v}\cdot\frac{\partial
f_{s}}{\partial{\bf r}}+\xi_{s}\left({\bf E}+{\bf v}\times{\bf
B}\right)\cdot\frac{\partial f_{s}}{\partial{\bf v}}=0$ (3) $\displaystyle{\bf
E}=-\left({\bf u}_{e}\times{\bf B}\right)-\frac{\nabla P_{e}}{n_{e}}+\eta{\bf
j}$ (4) $\displaystyle\frac{\partial{\bf B}}{\partial t}=-\nabla\times{\bf
E}\;\;\;;\;\;\;\nabla\times{\bf B}={\bf j}$ (5)

In previous equations, $f_{s}({\bf r},{\bf v},t)$ is the ion distribution
function (the subscript $s=p,\alpha$ refers to protons and $\alpha$ particles,
respectively), ${\bf E}({\bf r},t)$ and ${\bf B}({\bf r},t)$ are the electric
and magnetic fields, respectively, and $\xi_{s}$ is a constant reflecting the
charge to mass ratio of each species ($\xi_{p}=1$, $\xi_{\alpha}=1/2$). The
electron bulk speed is ${\bf u}_{e}=\left(\sum_{s}Z_{s}n_{s}{\bf u}_{s}-{\bf
j}\right)/n_{e}$, where $Z_{s}$ is the ion charge number ($Z_{p}=1$,
$Z_{\alpha}=2$), the ion density $n_{s}$ and the ion bulk velocity ${\bf
u}_{s}$ are the zero-th and first order moments of the ion distribution
function, respectively, and the electron density $n_{e}$ is evaluated through
the quasi-neutrality condition $n_{e}=\sum_{s}Z_{s}n_{s}$. Furthermore,
$P_{e}=n_{e}T_{e}$ is the electron pressure ($T_{e}$ is homogeneous and
stationary) and ${\bf j}=\nabla\times{\bf B}$ is the total current density.
The external small resistivity $\eta=2\times 10^{-2}$ is introduced to remove
spurious numerical instabilities. In Equations (3)–(5), masses, charges, time,
velocities and lengths are respectively normalized to the proton mass,
$m_{p}$, and charge, $e$, to the inverse of the proton cyclotron frequency,
$\Omega_{cp}^{-1}=m_{p}c/eB_{0}$ (being $c$ and $B_{0}$ the speed of light and
the ambient magnetic field), to the Alfvén velocity, $V_{A}=B_{0}/\sqrt{4\pi
n_{0,p}m_{p}}$ (being $n_{0,p}$ the equilibrium proton density), and to the
proton skin depth, $d_{p}=V_{A}/\Omega_{cp}$.

The described set of Equations (3)–(5) are solved in a $2.5D-3V$ phase space
domain, where $2.5D$ means that all vector components are retained but they
depend only on the two spatial coordinates $(x,y)$. The spatial domain, whose
size is $L=2\pi\times 20d_{p}$ in both directions, has been discretized with
$512^{2}$ grid-points and, at boundaries, periodic conditions are implemented.
Each direction of the velocity domain of each species (protons and $\alpha$
particles) is discretized with $71$ grid-points in the range
$v_{j,s}=\left[-v_{max,s},v_{max,s}\right]\;\;(j=x,y,z)$, where $v_{max,s}=\pm
5v_{th,s}$ (being $v_{th,s}=\sqrt{k_{B}T_{0,s}/m_{s}}$ the ion thermal speed,
with $T_{0,s}$ the equilibrium temperature). Boundary conditions in velocity
space assume $f_{s}(|v|>v_{max,s})=0$ in each direction. The time step $\Delta
t$ has been chosen in such a way that the CFL condition for the numerical
stability of the Vlasov algorithm is satisfied (Peyret & Taylor, 1986). To
control the numerical accuracy of the simulation, the basic set of
conservation laws of Vlasov equation is continuously monitored during the
simulations: relative variation of proton mass, $\alpha$ particle mass, total
energy and entropy are about $2.5\times 10^{-3}\%$, $3.5\times 10^{-1}\%$,
$0.6\%$ and $0.4\%$, respectively.

At the initial time of the simulation ($t=0$), both ion species have
Maxwellian distribution functions, with homogeneous densities, and an uniform
background out-of-plane magnetic field ${\bf B_{0}}=B_{0}{\bf e}_{z}$ is
imposed. In order to mimic the physical conditions of the pristine solar wind,
we assume that i) the $\alpha$ particle to proton density ratio
$n_{0,\alpha}/n_{0,p}=5\%$ and ii) we consider $\alpha$ particles, protons and
electrons at the same initial temperature $T_{0,\alpha}=T_{0,p}=T_{e}$. The
equilibrium is then perturbed through a $2D$ spectrum of Fourier modes in the
plane $(x,y)$, for both proton bulk velocity and magnetic field, such that
proton bulk velocity and magnetic field are globally uncorrelated. Energy is
injected with wave numbers in the range $0.1<k<0.3$ ($2\leq m\leq 6$, where
$k=2\pi m/L$) and random phases. We did not introduce neither density
disturbances nor parallel variances, i.e. $\delta n=\delta u_{z}=\delta
B_{z}=0$ at $t=0$. The r.m.s. level of fluctuations is $\delta B/B_{0}=1/3$,
while $\beta_{p}=2v_{th,p}^{2}/V_{A}^{2}=0.5$.

The kinetic evolution of protons and $\alpha$ particles is numerically
investigated in a decaying turbulence. During the simulation, nonlinear
couplings between fluctuations generate a turbulent cascade, where the
injected energy is transferred along the spectrum to smaller, kinetic scales.
The time $t^{*}$ of maximum turbulent activity can be identified as the peak
time of the spatially averaged out-of-plane current density, $\langle
j_{z}^{2}\rangle$, a standard indicator for small-scale magnetic gradients.
All the analysis has been performed using one snapshot of the simulation at
$t^{*}\Omega_{cp}=49$, when decaying turbulence approaches a quasi-steady
state. A more detailed description of the numerical results of the simulation
can be found in Valentini et al. (2016).

###  2.2 Numerical results

An overview of the differential kinetic dynamics of the ion species, namely
protons and $\alpha$ particles, associated to the development of the turbulent
cascade, is presented in Figure 1.

Figure 1: (Color online) 2D contour plots for the HVM multi-component
simulation at $t=t^{*}$. (a) The out-of-plane total current density $j_{z}$;
(b) the local energy transfer rate $\epsilon_{d_{p}}$; (c) ${\bf E}\cdot{\bf
j}$; (d) the proton temperature $T_{p}$; (e) the out-of-plane proton vorticity
$\omega_{z,p}$; (f) ${\bf b}\cdot\omega_{p}$; (g) the $\alpha$ particle
temperature $T_{\alpha}$; (h) the out-of-plane $\alpha$ particle vorticity
$\omega_{z,\alpha}$; and (i) ${\bf b}\cdot\omega_{\alpha}$.

Turbulence leads to the generation of small-scale coherent structures, such as
current sheets, filaments, or strongly shared flows, which can be recognized
in the two-dimensional patterns in physical space. In particular, panel (a) of
Figure 1 shows the shaded contours of the out-of-plane current density,
$j_{z}$. The current density becomes very intense in between magnetic islands,
where magnetic reconnection can locally occur.

Panel (b) shows the contour map of the local energy transfer rate,
$\epsilon_{l}(x,y)$, estimated using the velocity and magnetic field (in
Alfvén units) two-point increments at the scale $l=d_{p}$. Positive and
negative filaments of strong energy transfer $\epsilon_{l}$ are present, but
the spatial correlation with strong current sheets is unclear. There is a
shift between the current layers and this cascade-transfer indicator, probably
due to the fact that the former are the sites of dissipation, while the LET is
high at the very active nonlinear regions, such as the upstream parts of the
reconnecting current sheets.

In panel (c) we display the contour map of ${\bf E}\cdot{\bf j}$, a proxy of
energy conversion between fields and particles (Wan et al., 2015; Yang et al.,
2017). Large positive values are observed in the proximity of the strongest
current sheets, suggesting that the electromagnetic energy might be converted
into kinetic and thermal energy where magnetic reconnection is likely to
occur.

Panels (d) and (g) of Figure 1 show the maps of proton and $\alpha$-particle
temperature, respectively. Comparison with panel (a) highlights that the
presence of strong current sheets corresponds to temperature enhancement of
both ions. Although the initial conditions of the simulation infer equal and
uniform temperature for the two ion species, during the turbulent evolution of
the system both ions show a non-uniform temperature pattern, with similar
inhomogeneity as for the current $j_{z}$. Color bars were chosen so that the
initial temperature $T_{0}$ is white, while red and blue indicate local
heating and cooling, respectively. A definite temperature increase can be
observed close to the region of strong magnetic field gradients, meaning that
the heating processes are influenced by the topology of the magnetic field.
Preferential heating is also evident, as the $\alpha$ particles temperature is
nearly twice that of the protons. Moreover, while the protons display roughly
equal heating and cooling with respect to the initial temperature $T_{0}$, the
$\alpha$ particles show predominant temperature increase.

Enhanced ion heating was recently observed in spatial regions nearby the peaks
of the ion vorticity $\omega=\nabla\times{\bf v}$ (Servidio et al., 2015;
Franci et al., 2016; Valentini et al., 2016). Previous studies have shown the
association between viscous-like effects, velocity shears (Markovskii et al.,
2006; Del Sarto et al., 2016) and vorticity (Huba, 1996), including the
dependence on the sign of ${\bf B_{0}}\cdot{\bf\omega}$ (Parashar & Matthaeus,
2016). In particular, it has been shown that the vorticity may enhance local
kinetic effects, with a generalized resonance sign-dependence condition for
the enhancement or reduction of the ion heating and temperature anisotropy
(Parashar & Matthaeus, 2016). In order to explore such correlation, the maps
of the out-of-plane ion vorticities $\omega_{z,p}$ and $\omega_{z,\alpha}$,
are shown in panels (e) and (h) of Figure 1. A pattern of small-scale
vorticity structures can be observed, similar to the case of the current.
However, a more complex topology is observed for the $\alpha$ particles
vorticity, with the splitting of the structures in smaller filaments already
observed in Hall-MHD turbulence (Martin, 2013).

Finally, in panels (f) and (i) of Figure 1 we show the 2D contour plot of
${\bf b}\cdot{\bf\omega}$ for both protons and $\alpha$ particles,
respectively, where ${\bf b}={\bf B}-{\bf B_{0}}$ indicates the magnetic field
fluctuations. A behavior similar to $\omega_{z}$ is observed, so that both
positive $\omega_{z}$ and ${\bf b}\cdot{\bf\omega}$ correspond to increased
temperature, for both ion species.

Figure 2: Joint probability distributions for (a) $T_{p}$ and ${\bf
b}\cdot{\bf\omega}_{p}$; (b) $T_{\alpha}$ and ${\bf
b}\cdot{\bf\omega}_{\alpha}$; (c) $Q_{p}^{1/2}$ and $\xi_{p}$; (d)
$Q_{\alpha}^{1/2}$ and $\xi_{\alpha}$.

The overview of the numerical results of the HVM multi-species simulation
given in Figure 1 highlights a general enhancement of the ion temperature
related to the presence of thin current sheets. A more evident correlation is
found with the ion out-of-plane vorticity $\omega_{s}$ and with ${\bf
b}\cdot{\omega}_{s}$. To better appreciate the robustness of the observed
correlation between $T_{s}$ and ${\bf b}\cdot{\bf\omega}_{s}$, in panel (a) e
(b) of Figure 2 we show the average joint probability distributions
$P(T_{p},{\bf b}\cdot{\bf\omega}_{p})$ and $P(T_{\alpha},{\bf
b}\cdot{\bf\omega}_{\alpha})$, respectively. A good correlation is observed
for both ion species, that we quantify through the nonlinear Spearman
correlation coefficients, $C_{S}$, indicated in each panel of Figure 2.
Correlations are stronger for protons $[C_{S}(T_{p},{\bf
b}\cdot{\bf\omega}_{p})=0.76]$, than for $\alpha$ particles
$[C_{S}(T_{\alpha},{\bf b}\cdot{\bf\omega}_{\alpha})=0.53]$. However, the
correlation between the temperature and the out-of-plane vorticity (i.e., the
vorticity component parallel to the background magnetic field rather than to
the magnetic fluctuations) is larger for the $\alpha$ particles:
$C_{S}(T_{p},\omega_{z,p})=0.49$ and
$C_{S}(T_{\alpha},\omega_{z,\alpha})=0.63$ (joint distributions are not shown
for this case). The heating is thus better correlated with the out-of-plane
vorticity for the $\alpha$ particles, but with the vorticity component
parallel to the local magnetic fluctuations for the protons. This may be
related to the heavier $\alpha$ particles being slower than protons in
following the magnetic field fluctuations.

The heating mechanisms are also correlated with strong deformation of the ion
velocity distribution (Servidio et al., 2012; Valentini et al., 2016). For
each ion species and in each point of the simulation domain, the non-
Maxwellian measure can be defined as (Greco et al., 2012; Vasconez et al.,
2015; Pucci et al., 2016; Valentini et al., 2017; Pezzi et al., 2017c):

$\displaystyle\xi_{s}=\frac{1}{n_{s}}\sqrt{\int{(f_{s}-g_{s})^{2}d^{3}v}}.$
(6)

The above scalar function measures the deviation of the observed distribution
$f_{s}$ from the associated Maxwellian $g_{s}$, which is built using the local
density, bulk velocity and temperature. The latter quantity is also intimately
related to the so-called velocity space cascade, recently observed in space
plasmas (Servidio et al., 2017).

It is useful at this point to considered the agyrotropy of the pressure
tensor, i.e. its departure from cylindrical symmetry around the local magnetic
field direction. Non-gyrotropy is important as a known source of
instabilities, and may help identifying spatial structures in the plasma
(Scudder & Daughton, 2008; Aunai et al., 2013; Swisdak, 2016). For both ion
species, given the 3$\times$3 pressure tensor with elements $P_{ij}$ in an
arbitrary Cartesian system with coordinates $(x,y,z)$, the measure of
agyrotropy can be defined as the scalar (Swisdak, 2016):

$\displaystyle
Q_{s}=1-\frac{4I_{2,s}}{(I_{1,s}-P_{\parallel,s})(I_{1,s}+3P_{\parallel,s})}\
,$ (7)

with $I_{1}=P_{xx}+P_{yy}+P_{zz}$, and
$I_{2}=P_{xx}P_{yy}+P_{xx}P_{zz}+P_{yy}P_{zz}-(P_{xy}P_{yx}+P_{xz}P_{zx}+P_{yz}P_{zy})$.
$P_{\parallel}$ here represents the parallel pressure, where we have omitted
the subscript $s$ for simplicity. A good correlation is observed between the
non-gyrotropy of the ion pressure tensor, $Q_{s}^{1/2}$, and the non-
Maxwellianity of the ion velocity, $\xi_{s}$, as shown in the joint
probability distribution maps in panels (c) and (d) of Figure 2 for protons
and $\alpha$ particles, respectively. This mechanism works better for $\alpha$
particles $[C_{S}(Q_{\alpha}^{1/2},\xi_{\alpha})=0.72]$ with respect to
protons $[C_{S}(Q_{p}^{1/2},\xi_{p})=0.48]$, since deviations from Maxwellian
are significantly stronger for the $\alpha$ particles than for protons.

Table 1 collects the Spearman correlation coefficients estimated for pairs of
quantities for which $C_{S}\gtrsim 0.4$, namely if at least a significant
moderate correlation is present. These also include the temperature anisotropy
$A_{s}=T_{\perp,s}/T_{\parallel,s}$, defined for both ion species, an
important diagnostic for kinetic processes and instabilities. In the Table, we
have omitted the weakly correlated pairs, for which $C_{S}\lesssim 0.4$.

Spearman Correlation Coefficients
---
$(T_{p},\omega_{z,p})=0.49$ |  | $(T_{p},{\bf b}\cdot{\bf\omega}_{p})=0.76$ |  | $(T_{p},A_{p})=0.42$
$(T_{\alpha},\omega_{z,p})=0.63$ |  | $(T_{\alpha},{\bf b}\cdot{\bf\omega}_{\alpha})=0.53$ |  | $(T_{\alpha},A_{\alpha})=0.52$
$(A_{p},\omega_{z,p})=0.45$ |  | $(A_{p},{\bf b}\cdot{\bf\omega}_{p})=0.38$ |  | $(Q_{p}^{1/2},\xi_{p})=0.48$
$(A_{\alpha},\omega_{z,\alpha})=0.57$ |  | $(A_{\alpha},{\bf b}\cdot{\bf\omega}_{\alpha})=0.52$ |  | $(Q_{\alpha}^{1/2},\xi_{\alpha})=0.72$
$(\omega_{z,\alpha},\omega_{z,p})=0.65$ |  | $({\bf b}\cdot{\bf\omega}_{\alpha},{\bf b}\cdot{\bf\omega}_{p})=0.69$ |  | $(\xi_{z,\alpha},\xi_{z,p})=0.61$
Table 1: Nonlinear Spearman correlation coefficients, $C_{S}$, between pairs
of quantities for which $C_{S}\gtrsim 0.4$. All quantities are estimated using
the HMV simulation results at the time $t=t^{*}$.

To summarize the results of the above analysis, larger values of the
deformation of the ion velocity and of the pressure tensor agirotropy are
recovered in spatial positions nearby the peaks of the ion vorticity and close
to peaks of the current density. In these regions, differential heating and
temperature anisotropy are also observed. Ion heating and kinetic effects are
in general more strongly correlated with the vorticity than with the current
density profile.

##  3 Local energy transfer rate in Vlasov-Maxwell simulations

###  3.1 Properties of the local energy transfer rate

Incompressible and isotropic MHD turbulence in plasmas can be described by the
exact relation for the energy flux through the scales (see Equation 1). This
Yaglom-like scaling law has been extensively tested in the solar wind,
recovering very good agreements. The use of HVM multi-component simulation,
described in Section 2, can provide a better understanding of the connection
between the MHD turbulent cascade and the relative kinetic effects on the ion
velocity distributions since the ‘in situ’ solar-wind measurements have
several limitations.

In Figure 3, we present the validation of the PP law for the turbulent Vlasov
simulation used in this work, as estimated by averaging Equation (2). In the
range of scales roughly corresponding to the inertial range of MHD turbulence,
i.e. for $2d_{p}\lesssim l\lesssim 10d_{p}$, the scaling of $-Y(l)$ is
compatible with a linear law (Figure 3). This suggests that the turbulent
energy cascade is well developed in such range, at the time $t^{*}$ of the
simulation. However, the limited extension of the inertial range does not
allow a more precise and quantitative estimate (Parashar et al., 2015). It
should be pointed out that the law as defined in Equation (1), without
absolute value, is indicating the direction of the cascade towards small
scales. The limited extension of the range observed in the HVM simulation may
arise from several facts, which are physically meaningful. First, the system
described here is size-limited due to computational constraints: the
correlation length is limited to roughly 10$d_{p}$ —relatively short as
compared to the several decades of MHD turbulence commonly observed in the
solar wind. Second, the PP law is exact only under the assumption of full
isotropy, while our 2D simulation has a mean magnetic field. Third, the ion
skin depth is a characteristic length of the system, which breaks the validity
of the MHD law. A more precise description could be obtained in terms of the
Hall MHD version of the PP law (Andrés et al., 2018). This development is left
for a future work. Fourth, in the derivation of these MHD-based scaling laws,
isotropy of the pressure tensor is assumed, which does not hold in Vlasov
turbulence. Finally, processes such as ion-cyclotron resonances and Landau
damping might transfer free energy from the physical space to the velocity
space, producing non-Maxwellian effects and, eventually, heating. All these
caveats suggest that the analysis in terms of PP law and LET can only be
meaningful for increments in the inertial range, $l>d_{p}$.

Figure 3: (Color online) The Yaglom law validation for turbulent Vlasov
simulation (red markers and dashed line). The averaged scaling
$Y=(Y^{+}+Y^{-})/2$ is shown. The black dotted line represents an example of
linear scaling, and is shown for reference only.

The local proxy $\epsilon_{l}(x,y)$ has been evaluated at different spatial
scales $l$ using the HVM simulation results. In Figure 4 we show the scalogram
of the LET $\epsilon_{l}$ along a one-dimensional cut of the 2D spatial domain
of the simulation. Note that a similar analysis has been performed for the PVI
scalograms (Greco et al., 2016). Here we consider a diagonal path $r$,
normalized to $d_{p}$, that crosses the simulation box several times. The
channels of energy being transferred across scales are clearly visible, and
reveal the complex, interwoven pattern of positive and negative flux that
determines the neat energy flux (Coburn et al., 2014; Sorriso-Valvo et al.,
2018).

Figure 4: (Color online) The scalogram of LET $\epsilon_{l}$, plotted as color
map at different scales $l/d_{p}$, estimated at different positions $r/d_{p}$
along a one-dimensional cut of the simulation domain. The alternate positive
and negative energy transfer regions are evident.

In order to characterize the statistical properties of the local energy
transfer rate, in Figure 5, we show the probability distribution functions
(PDFs) of the negative (left panel) and positive (central panel) standardized
LET values (i.e., $\epsilon_{l}(x,y)$ has been rescaled to have zero mean and
unit standard deviation), for three different scales, namely $l=d_{p}$ (blue
diamonds), $4d_{p}$ (green triangles) and $8d_{p}$ (red circles). In agreement
with the PP law, for a given scale the positive tails are considerably higher
and shallower than the negative ones, indicating the neat energy flux
corresponding to the direct turbulent energy cascade towards small scales.
This is clearly visible from the PDFs represented in Figure 5.

Both the positive and negative PDFs behave roughly as stretched exponential
functions $\sim\exp{(-b\epsilon_{l}^{c})}$, where $c$ is the parameter that
controls the shape of the distribution tails. In particular, $c=2$ indicates a
Gaussian tail, $c=1$ an exponential tail, and for $c\to 0$ the tail approaches
a power-law. The curvature parameter $c$, obtained from the fit at each scale,
is shown in the right panel of Figure 5. Starting from the exponential value
$c\sim 1$ at large scales (say for $l\gtrsim 10d_{p}$), its value decreases
with the scales, indicating the formation of highly localized small-scale
structures, typical of intermittency. The two tails of the PDFs have different
scaling, so that the positive tail has a more evident, faster decay than the
negative tail. Such difference quantitatively confirms the imbalance already
observed through the PDF shape. In the range corresponding to the MHD inertial
range, the scaling of $c$ for the positive tail is well described by a power-
law $c^{\gamma}_{\pm}$, with exponent $\gamma_{+}\sim 0.92\pm 0.06$. For the
negative tail, the scaling is less evident, but can be considered compatible
with a power law with exponent $\gamma_{-}\sim 0.37\pm 0.13$. This confirms
that the numerical simulation has reached a stage of fully developed
turbulence, with a strongly intermittent energy cascade, whose characteristics
are well captured by the LET. Note that a similar behavior has been recently
observed in solar wind measurements (Sorriso-Valvo et al., 2018). At smaller
scales, for $l\lesssim d_{p}$, the parameter seems to reach a limiting value
for both the positive and negative tails, indicating the saturation of
intermittency. In this short range of scales, the positive tail of the PDF is
described by $c\sim 0.1$, closer to power-law decay. The negative tail
converges towards a slightly larger value $c\sim 0.3$, indicating a faster
than power-law decay, and confirming once more the quantitative difference
between positive and negative energy transfer rate.

Figure 5: (Color online) In the left panel, the probability distribution
functions (PDFs) of the standardized, negative $\epsilon_{l}(x,y)$ are shown
for three different scales, namely $l=d_{p}$ (blue diamonds), $4d_{p}$ (green
triangles) and $8d_{p}$ (red circles), using logarithmic axes. In the central
panel, the PDFs of positive $\epsilon_{l}(x,y)$ are shown for the same scales,
with the same symbols. In both panels, the different lines are the stretched
exponential fits, obtained through a standard least-$\chi^{2}$ procedure. The
right panel shows the scaling properties of the curvature parameter $c(l)$,
for the positive (violet squares) and negative (orange circles) tails of the
PDFs. The inertial range power-law fits are also indicated as dashed lines,
the scaling exponents being respectively $\gamma_{+}=0.92$ and
$\gamma_{-}=0.37$.

###  3.2 Conditioned analysis of kinetic quantities

After establishing the statistical properties of the LET in our hybrid
simulation, we have performed the conditioned average analysis proposed by
Osman et al. (2012), and recently performed using the LET (Sorriso-Valvo et
al., 2018). The conditioned analysis is designed to highlight the existence of
robust features in the data that might be hidden by the turbulent
fluctuations. To this goal, we have selected different values of threshold
$\theta_{d_{p}}$ of the LET $\epsilon_{d_{p}}$, estimated at the bottom of the
inertial range $l\simeq 2d_{p}$, to identify locations associated with large
energy flux at the end of the fluid-like nonlinear turbulent cascade. Then,
for each threshold, some of the variables described in Section 2 have been
averaged only in those locations where the LET overcomes the threshold:
$\epsilon_{d_{p}}\geq\theta_{d_{p}}$ for positive thresholds (cascade towards
smaller scales), and $\epsilon_{d_{p}}\leq\theta_{d_{p}}$ for negative
thresholds (cascade towards higher scales). This gives the conditioned
averaged values, for example $\langle T_{p}|\theta_{d_{p}}\rangle$ in the case
of the proton temperature.

Through the above procedure, the turbulent fluctuations are all averaged to
their mean value, except where systematic variations are present. The same
procedure is repeated using all the points located at a given distances
$\delta$ from the location of large LET, and for different values of $\delta$.
This provides the average profile of the physical quantities in the vicinity
of the bursts of energy flux, as a function of the distance from the LET
burst. By repeating the same analysis using different thresholds, it is then
possible to determine the role of the energy flux in the robust modification
of the physical quantities.

Figure 6: (Color online) Conditioned averaged quantities as a function of the
distance from the peak of the LET, estimated at the scale $l\simeq 2d_{p}$,
for four positive and four negative LET threshold values $\theta_{d_{p}}$
(coded in different line colors, as indicated at the bottom of the Figure).
Each panel refers to one of the physical quantities (see Section 2, given in
the legend located at the bottom of the panel). All curves have been
vertically shifted for clarity, and are thus presented in arbitrary units. In
each panel, the curves for larger positive thresholds (full lines) are those
shown at upper positions, and the larger negative thresholds (dashed lines)
are shown at lower positions. The dotted horizontal line has the sole purpose
of separating the curves relative to positive (above the line) and negative
(below the line) thresholds.

The 12 panels of Figure 6 show the conditioned average profiles obtained using
the HVM simulation results, for some of the physical quantities described in
Section 2 and indicated in each panel. The average has been computed up to a
distance $\delta=12d_{p}$ from the LET peaks. For each panel, the different
line styles refer to the different threshold values, as listed at the bottom
of the figure. All curves have been vertically shifted for clarity, and are
therefore given in arbitrary units. The dotted horizontal line has the only
purpose of separating the positive from the negative threshold values. The
figure shows that most of the variables have a clear enhancement at $\delta=0$
in the positive LET peak cases, which is more evident for larger threshold
values. At the same time, in most cases the negative LET peaks are associated
with a reduction of the conditioned average, again more evident for larger
negative thresholds. This observation generally suggests that kinetic
processes are more effective within a few $d_{p}$ from the intermittent
structures, or more precisely from the locations where the fluid-scale
turbulent cascade has produced a larger flux of energy.

A more detailed look at the individual panels of Figure 6 can provide
information that may help uncovering the kinetic processes occurring in the
simulation. Panel A shows that the off-plane current $j_{z}$ is not well co-
located with the positive LET structures, as the conditioned average profiles
show a weak peak near $\delta\simeq 3d_{p}$. On the contrary, regions of
strong negative LET are associated to smaller current, as evident from the
bottom part of panel A. This could arise from the fact that negative LET
structures are associated with regions where the turbulent production of small
scales is locally inhibited, indicating possible equilibrium-like patterns,
that results from a process of suppression of nonlinearity (Servidio et al.,
2008).

In panels B and C, the average proton and $\alpha$ particle vorticity peak at
large positive LET locations, and, as for the current, is minimal for large
negative LET. The important correlation between vorticity structures and
kinetic features has been already observed in particle-in-cell simulations
(Parashar & Matthaeus, 2016).

Panel D shows that the exchange of energy between particles and fields,
represented here through the quantity $\mathbf{E}\cdot\mathbf{j}$, is weakly
enhanced near the LET positive peaks, but is relatively unaffected by the
presence of negative LET peaks. This is in good agreement with previous works,
where it has been shown that locally the cascade produces narrow heating and
dissipation patterns (Karimabadi et al., 2013).

Panels E and F display enhanced temperature near the positive LET peaks for
both species, which extends up $\delta\sim$ 2–3$d_{p}$ from the peaks. For the
$\alpha$ particles, there is indication of colder temperature near the large
negative LET points, which highlights the possible presence of a differential
heating processes of the two species (Kasper et al., 2008; Maruca et al.,
2012).

Panels G and H show the conditioned profiles of the temperature anisotropy
$A_{s}$. In panel G, the proton temperature anisotropy is only weakly affected
by large positive energy transfer, while a general anisotropy increase can be
observed in correspondence with large negative LET. On the contrary, as can be
seen in panel H, the $\alpha$ particle temperature anisotropy increases
(decrease) near the large positive (negative) LET structures. In this case,
the LET dependence is similar for the temperature anisotropy and for the total
temperature. This suggests that the presence of large turbulent energy
transfer increases predominantly the perpendicular temperature, indicating for
example the possible role of ion-cyclotron or mirror instabilities (Matteini
et al., 2013).

Panels I and J show that the quantity ${\mathbf{b}}\cdot\omega_{s}$ has
similar conditioned averages as $\omega_{zs}$, for both protons and $\alpha$
particles.

Finally, panels K and L show that the parameter $\xi_{s}$, quantifying the
general deviation from Maxwellian VDF, is clearly larger at both positive and
negative LET structures, and for both ion species. This last observation
confirms that the kinetic processes resulting in modification of the ion VDFs
are predominantly seen in the proximity of the locations with enhanced
turbulent energy flux, where free energy is available for small-scale
processes to occur. Additionally, it suggests that regions characterized by
absent or reduced direct energy transfer towards small scales (large negative
LET) are also characterized by the presence of robustly non-Maxwellian VDF.
Whether this indicates that kinetic effect may provide feedback to the larger
scales through some form of inverse cascade remains an open question that
deserves further analysis.

It is interesting but not surprising to note that, for most of the physical
quantities studied here, the observed peaks extend to approximately 2–3$d_{p}$
around the LET structures. This indicates that the effects of the kinetic
features are typically smeared out within such range.

##  4 Conclusions

Exploring the details of the cross-scale connections between the MHD turbulent
energy cascade and the small-scale kinetic processes is an important key to
understand the role of dissipation mechanisms in collisionless plasmas
(Schwartz et al., 2011; Alexandrova et al., 2013; Chen, 2016). Numerical
simulations can provide important insight on general processes, in particular
when experimental systems and observations are not easily controlled.

In this article, we have presented the analysis of one snapshot in the quasi-
steady state turbulence of a two-dimensional Hybrid Vlasov-Maxwell numerical
simulation (HVM), with the inclusion of $\alpha$ particles. The range of
scales that have been investigated includes both the fluid range, where MHD
turbulence transfers energy across scales, and the sub-ion range, where
kinetic effects dominate the dynamics. This multi-scale investigation allows
to explore the role of the turbulent energy that reaches the typical ion
scales in the form of highly localized structures and the kinetic processes
occurring at small scales. The former is represented by means of a proxy of
the Local Energy Transfer rate (LET), whose heuristic definition is based on
the linear scaling laws of the mixed third-order moment of the MHD
fluctuations. The latter are described here through several quantities that
can be related to kinetic processes, as for example the ion temperature and
its anisotropy with respect to the magnetic field direction, or an index of
deviation from Maxwellian particle velocity distribution.

We have initially described the correlations between such kinetic quantities,
first by visually comparing their maps, then by looking at some examples of
joint probability distributions of such quantities pairs, and finally by
estimating the nonlinear correlation coefficients between all the pairs.
Several diagnostics show significant correlation. For example, the ion
vorticity and the associated temperature show moderate correlation for both
species, confirming previous observations that the temperature is enhanced in
correspondence of vorticity structures (Servidio et al., 2015; Franci et al.,
2016; Valentini et al., 2016; Parashar & Matthaeus, 2016). The deviation from
Maxwellian (Servidio et al., 2012; Valentini et al., 2016) is also very well
correlated with the pressure tensor anisotropy, revealing its relevant
contribution in determining such deviation.

We have then analyzed the statistical properties of the energy transfer rate
in HVM turbulence. The domain-averaged third-order moment suggests that the
turbulent cascade might be fully developed in the range approximately between
$\sim 2d_{p}$ and $\sim 10d_{p}$, where the measured energy flux is compatible
with linear scaling. A direct cascade of energy is present (the negative sign
of the third-order moment $Y$), in agreement with the expectations for
turbulence in the MHD inertial range of scales. However, looking at the space-
scale distribution of the local contributions to such scaling law, it is clear
that both positive and negative energy fluxes exist, and are in comparable
quantity, all across the simulation domain, the negative sign being the global
balance between those.

The scaling properties of the LET were modeled in terms of stretched
exponential distributions, which provided a satisfactory description of the
PDFs at all scales. The imbalance between positive and negative values of
$\epsilon$ are well described by the asymmetry of their distributions.
Moreover, the shape parameter of the fitted distributions shows power-law
scaling in the inertial range, in agreement with previous observations of
solar wind turbulence (Sorriso-Valvo et al., 2018) and confirming the
reliability of the proxy for the description of the turbulence properties.

Finally, we have explored the presence of correlations between the LET and the
quantities described before. To this aim, we have calculated the conditionally
averaged profile of each of those quantity, around the locations where the LET
exceeds some given thresholds (Osman et al., 2012; Sorriso-Valvo et al.,
2018). This analysis may provide more detailed information that can help
understand which kinetic processes are responsible for the removal (or
transformation) of the turbulent energy from the system. These profiles shows
that most of the physical quantities related to “dissipative” or kinetic
processes are enhanced near regions of large positive LET, indicating a large
flux of energy towards small scales. Conversely, these are often reduced near
strong negative LET, likely associated to energy being transferred towards
large scales, and thus indicating less availability of energy in the kinetic
range.

Our results suggest, for example, that peaks of turbulent energy transfer are
associated with enhanced temperature. More in particular, the perpendicular
temperature increases near the high LET locations, indicating the possible
role of ion-cyclotron or mirror instabilities (Matteini et al., 2013). The
typical size of the region where the enhancement is observed is, for most
quantities, of the order of 2–3$d_{p}$, while at larger distance their
persistent fluctuations are averaged out. It is also clear from the analysis
that the vorticity is a better tracer of energy availability than the current
density (Servidio et al., 2015; Franci et al., 2016; Parashar & Matthaeus,
2016; Valentini et al., 2016), and that correlations are more evident for the
$\alpha$ particles than for protons (Valentini et al., 2016). Finally,
deviations from Maxwellian VDF are evident for both positive and negative
peaks of LET, an interesting feature that deserves further analysis.

The study of the local properties of the turbulent cascade can be a valuable
tool for the exploration of the kinetic range processes in space plasmas. The
validation of its use in numerical simulations provided in this work suggests
its possible use with the recently available high-quality plasma measurements
in the solar wind and in the magnetosphere, provided by the Multiscale
Magnetospheric mission MMS (Burch et al., 2016a). Extension to more complete
descriptions of the energy transfer rate could be also used, allowing, for
example, to examine the actual role played by compressive effects (Carbone,
2009; Andrés et al., 2018).

## Acknowledgements

This work has been partly supported by the Agenzia Spaziale Italiana under the
Contract No. ASI-INAF 2015-039-R.O “Missione M4 di ESA: Partecipazione
Italiana alla fase di assessment della missione THOR” and by the International
Space Science Institute (ISSI) in the framework of International Team 504
entitled “Current Sheets, Turbulence, Structures and Particle Acceleration in
the Heliosphere”. L.S.-V., D.P., S.S. and I.Z. acknowledge support from the
Faculty of the European Space Astronomy Centre (ESAC). Numerical simulations
here discussed have been run on the Newton parallel machine at the University
of Calabria (Italy).

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