2022

[1]Simone Benella

[1]INAF - Istituto di Astrofisica e Planetologia Spaziali, Roma, Italy

2]Dipartimento di Fisica, Università degli Studi di Roma Tor Vergata, Roma,
Italy

3]Swedish Institute for Space Physics, Uppsala, Sweden

# Kramers-Moyal analysis of interplanetary magnetic field fluctuations at sub-
ion scales

<EMAIL_ADDRESS>Mirko Stumpo<EMAIL_ADDRESS>Giuseppe Consolini
<EMAIL_ADDRESS>Tommaso Alberti<EMAIL_ADDRESS>Monica
Laurenza<EMAIL_ADDRESS>Emiliya Yordanova<EMAIL_ADDRESS>*  [  [

###### Abstract

In the framework of statistical time series analysis of complex dynamics we
present a multiscale characterization of solar wind turbulence in the near-
Earth environment. The data analysis, based on the Markov-process theory, is
meant to estimate the Kramers-Moyal coefficients associated with the measured
magnetic field fluctuations. In fact, when the scale-to-scale dynamics can be
successfully described as a Markov process, first- and second-order Kramers-
Moyal coefficients provide a complete description of the dynamics in terms of
Langevin stochastic process. The analysis is carried out by using high-
resolution magnetic field measurements gathered by Cluster during a fast solar
wind period on January 20, 2007. This analysis extends recent findings in the
near-Sun environment with the aim of testing the universality of the Markovian
nature of the magnetic field fluctuations in the sub-ion/kinetic domain.

###### keywords:

solar wind turbulence – Cluster – interplanetary magnetic field – Markov
processes

##  1 Introduction

During last decades, high-resolution magnetic field measurements gathered in
the inner Heliosphere have increased the interest in investigating the nature
of the magnetic field fluctuations observed at scales below the ion inertial
length Bruno16 . Observations gathered in the near-Earth environment were
recently accompanied by high-resolution observations in the vicinity of the
Sun, thus providing a comprehensive picture of the principal solar wind
properties as well as their radial evolution chen2020evolution ; Alberti20 .
For interplanetary magnetic field (IMF) fluctuations observed at scales below
0.1-1 Hz, the physical phenomena show the universal features of fully
developed turbulence with an important role played by the coupling between
magnetic and velocity fields near the Sun and a fluid-like turbulence above
$\sim 0.6$ astronomical units. All these features are consistent with results
obtained in the framework of fluid-like approximation of the solar wind
turbulence from the integral scale of the energy injection to approximately
$10\,d_{i}$, where $d_{i}$ is the ion inertial length Bruno16 . As usually
called in the literature leamon1998observational ; kiyani2009global ;
chhiber2021subproton  from here on we will refer to this large scale interval
characterized by a near-Kolmogorov spectrum as the inertial range. Conversely,
there is no general consensus on what are the mechanisms responsible for the
IMF fluctuations observed in the range below $d_{i}$, namely sub-ion/kinetic
range. As a matter of fact, several mechanisms acting at sub-ion/kinetic
scales have been proposed, such as wave-like fluctuations (e.g., Alfvén ion
cyclotron, kinetic Alfvén waves, whistler waves), coherent magnetic structures
(e.g., Alfvén vortices, current structures), magnetic reconnection processes
boldyrev2013toward ; lion2016coherent ; gary2009 ; Schekochihin_2009 ;
alexandrova2006 ; cerri2017reconnection . Moreover, alternative approaches
based on non-equilibrium statistical mechanics can also make important
contributions in unveiling some characteristics of processes occurring at sub-
ion/kinetic scales Carbone_2022 . An interesting method for investigating
complex time series within the framework of stochastic processes has been
developed and widely employed to characterize the energy transfer across the
turbulent cascade (pedrizzetti1994markov, ; friedrich1997description, ;
friedrich1997statistical, ; renner2001experimental, ; peinke2019fokker, ).
These authors suggest that the turbulent cascade can be described as a Markov
process whose two-points transition probabilities allow one to reproduce the
evolution of the Probability Distribution Functions (PDFs) of the longitudinal
velocity increments $u_{\ell}$ associated with the redistribution of energy
across scales. More specifically, the amplitude of $u_{\ell}$ at subsequent
scales within the inertial range can be represented as a Langevin process with
a drift force and a diffusion strength that depend on the scale. In the field
of space plasma turbulence, Strumik and Macek (strumik2008statistical, ;
strumik2008testing, ) shown that radial IMF fluctuations at typical time
scales within the inertial range appear to satisfy the Markov property and the
dynamics is accurately reproduced by means of the Fokker-Planck (FP) equation.

An important result by Kiyani et al. kiyani2009global  using ESA-Cluster
mission data, pointed out the existence of a global scale-invariance at scales
below the ion-inertial length, thus suggesting an intriguing difference with
respect to the anomalous scaling (i.e., the deviation from the linear scaling
predicted by the Kolmogorov theory, Kolmogorov41, ), which is ubiquitous in
fully developed turbulence, being the fingerprint of intermittency. Analogous
results were reported also in the case of the near-Sun environment through the
high-resolution observations gathered by the Parker Solar Probe mission during
the perihelia (e.g., see chhiber2021subproton, , and references therein).
Regarding the Markov property in the sub-ion/kinetic range, Benella et al.
(Benella_2022, ) recently showed that IMF fluctuations at these scales keep
satisfy such condition in the near-Sun environment, thus sharing with the
inertial range the local structure of the energy transfer across scales.
However, the two regimes appear to be independent in a statistical sense, i.e.
there is no correspondence between the IMF fluctuations measured in the
inertial range and the corresponding fluctuations observed in the sub-
ion/kinetic domain, thus supporting the idea that they are originated by
different physical mechanisms. Since in this analysis we use a stochastic
framework, it does not allow to discern between the different physical
processes involved at sub-ion/kinetic scales, but only to characterize and to
model the stochastic nature of the energy transfer, in terms of fluctuations,
through different scales. This energy transfer is described in terms of an
advection-diffusion stochastic equation through the scales. In fact, it has
been shown that the evolution of the statistics of the IMF fluctuations within
the sub-ion/kinetic range can be modeled through the FP equation. In this
case, the observed global-scale invariance can be viewed as the stationary
solution of the equation that governs the evolution in scale of the PDFs of
the rescaled IMF fluctuations (i.e., IMF fluctuations divided by their
standard deviations).

The aims of this work are two:

  * •

The first aim is to extend the analysis proposed in (Benella_2022, ), here
referred as Kramers-Moyal (KM) analysis, to high-frequency near-Earth
observations of solar wind gathered by the ESA-Cluster mission. In fact, the
comparison between results obtained in different solar wind conditions at
different heliocentric distances is crucial in order to look for an
universality of these statistical properties at the sub-ion/kinetic scales;

  * •

The second aim is to provide a general method for estimating the master curve
of the rescaled IMF PDFs in the sub-ion/kinetic range by only using the KM
coefficients evaluated on the IMF timeseries.

##  2 Data and Methods

###  2.1 Data

The data we used in this study were gathered by the ESA-Cluster B spacecraft
on 20 January 2007 in a 75 minutes time interval from 12:00 UT to 13:15 UT
with a sampling rate of $\sim 450$ samples/s. This period was characterized by
a fast solar wind stream ($v\sim 600$ km-1) with a mean interplanetary
magnetic field (IMF) of $B_{0}\sim 4$ nT and a plasma density of $n\sim 2$
cm-3. Since we are interested in IMF fluctuations parallel and perpendicular
with respect to the mean field, we rotated the IMF components in the minimum
variance reference frame. Therefore, we focus on the minimum ($B_{1}$) and
maximum ($B_{3}$) variance components, which are mainly aligned along
directions parallel and perpendicular to the mean field, respectively. The IMF
components are reported in Figure 1 (left panel) along with their power
spectral densities (PSDs; right panel). These PSDs have an $f^{-\beta}$
behavior with two distinct regimes separated by a spectral break located
around the ion-frequency $f_{i}\simeq 0.6$ Hz. For frequencies below $f_{i}$
we observe that the PSD exhibit a scaling à la Kolmogorov (i.e.,
$\beta\sim-5/3$), whereas for frequencies larger than $f_{i}$ the spectral
trend present the usual sub-ion/kinetic exponent $\beta\in[-7/3,-8/3]$
(Bruno16, ; leamon1998observational, ; kiyani2009global, ). In the following
we perform the analysis in the time domain by assuming that we are exploring
spatial scales $r$ via Taylor’s hypothesis, i.e., $r=\tau\,U_{0}$, being
$U_{0}$ the average plasma bulk speed.

###  2.2 Methods

The method we use in this work is based on the Markov process theory. By
introducing the increments of the IMF $b_{i}(\tau)\doteq B(t+\tau)-B(t)$
(where $i=\\{1,2,3\\}$), we can interpret $b_{i}(\tau)$ as a stochastic
processes that evolve across the time scales $\tau$. If the probability of
observing the increment $b_{1,i}$ at the scale $\tau_{1}$ given the increments
$b_{2,i}$ at the scale $\tau_{2}$ until $b_{\tau_{n},i}$ at the scale
$\tau_{n}$, with $\tau_{1}<\tau_{2}<\dots<\tau_{n}$ does not depend on
increments at scales larger than $\tau_{3}$ the process satisfy the Markov
condition, i.e.,

$p(b_{1,i},\tau_{1}\rvert b_{2,i},\tau_{2}\mathchar
24635\relax\;\dots\mathchar
24635\relax\;b_{n,i},\tau_{n})=p(b_{1,i},\tau_{1}\rvert b_{2,i},\tau_{2}).$
(1)

For a Markov process, the transition probability between $b_{1,i}$ and
$b_{3,i}$, i.e., the probability of observing the fluctuation $b_{1,i}$ at
scale $\tau_{1}$ given $b_{3,i}$ at scale $\tau_{3}$, can be written in terms
of the Chapman-Kolmogorov (CK) equation by integrating over fluctuations
$b_{2,i}$ at the intermediate scales $\tau_{2}$ such that
$\tau_{1}<\tau_{2}<\tau_{3}$ (risken1996fokker, ),

$p(b_{1,i},\tau_{1}\rvert
b_{3,i},\tau_{3})=\int_{-\infty}^{+\infty}p(b_{1,i},\tau_{1}\rvert
b_{2,i},\tau_{2})p(b_{2,i},\tau_{2}\rvert b_{3,i},\tau_{3})db_{2,i}.$ (2)

The time evolution of the transition probability is governed by the master
equation

$-\frac{\partial}{\partial\tau}p(b_{i},\tau\rvert
b_{i}^{\prime},\tau^{\prime})=\mathcal{L}_{KM}(b_{i},\tau)p(b_{i},\tau\rvert
b_{i}^{\prime},\tau^{\prime}),$ (3)

where the operator $\mathcal{L}_{KM}(b_{i})$ is the Kramers-Moyal (KM)
operator and it is defined as

$\mathcal{L}_{KM}(b_{i},\tau)=\sum_{k=1}^{\infty}\biggl{(}-\frac{\partial}{\partial
b_{i}}\biggr{)}^{k}D^{(k)}(b_{i},\tau).$ (4)

The functions $D^{(k)}(b_{i},\tau)$ are called KM coefficients and are defined
as

$D^{(k)}(b_{i},\tau)=\frac{1}{k!}\,\lim_{\delta\to
0}\,\frac{1}{\delta}\,\mathbb{E}[(b_{i}(\tau-\delta)-b_{i}(\tau))^{k}\rvert
b_{i}(\tau)]$ (5)

as also introduced in an early work by Kolmogorov kolmogorov1931analytical .
The minus sign on the l.h.s. of Equation (3) is due to the direction of the
time evolution from larger towards smaller scales (renner2001experimental, ).
In principle, the KM expansion encloses an infinite number of terms that
contribute to the time evolution of the statistics of the process. An
important class of stochastic processes is represented by those for which the
Pawula’s theorem is valid. In fact, the Pawula’s theorem states that if
$D^{(k)}(b_{i},\tau)=0$, then all the coefficients of order $k\geq 3$ vanish
and the KM expansion reduces to the Fokker-Planck (FP) equation
(risken1996fokker, )

Figure 1: Left: IMF minimum (blue) and maximum (red) variance components on 20
January 2007 between 12:00 and 13:15. Right: PSD of $B_{1}$ (blue) and $B_{3}$
(red). Black dotted lines are $-5/3$ and $-8/3$ slopes, whereas the red dotted
line indicates the ion-frequency $f_{i}$.

$-\frac{\partial}{\partial\tau}p(b_{i},\tau\rvert
b_{i}^{\prime},\tau^{\prime})=\biggl{[}-\frac{\partial}{\partial
b_{i}}D^{(1)}(b_{i},\tau)+\frac{\partial^{2}}{\partial
b_{i}^{2}}D^{(2)}(b_{i},\tau)\biggr{]}p(b_{i},\tau\rvert
b_{i}^{\prime},\tau^{\prime}).$ (6)

The first-order KM coefficient $D^{(1)}(b_{i},\tau)$ is representative of the
drift term, whereas the second-order KM coefficient $D^{(2)}(b_{i},\tau)$ is
related to the diffusion strength, which modulates the amplitude of the delta-
correlated Gaussian noise, $\Gamma(\tau)$, in the corresponding Langevin
equation

$-\frac{\partial
b_{i}}{\partial\tau}=D^{(1)}(b_{i},\tau)+\sqrt{2D^{(2)}(b_{i},\tau)}\,\Gamma(\tau).$
(7)

In this Langevin-like description of IMF fluctuations, the diffusion
coefficient accounts for the stochastic character of the values assumed by the
fluctuations at different scales. Since the true values of KM coefficients are
not accessible from data, we will approximate them as their finite-time
version for a sufficient small value of $\delta$

$D^{(k)}_{\delta}(b_{i},\tau)=\frac{1}{k!\delta}\,\mathbb{E}[(b_{i}(\tau-\delta)-b_{i}(\tau))^{k}\rvert
b_{i}(\tau)].$ (8)

Here, we applied the above technique to the small-scale increments of the
magnetic field at the sub-ion/kinetic scales, starting from the verification
of the CK Equation and successively evaluating the KM coefficients up to the
fourth order.

##  3 Results

The first step of the analysis is to check the Markov property of $B_{1}$ and
$B_{3}$ fluctuations at sub-ion/kinetic scales. By inspecting the PSD we
observe that the instrumental noise affects magnetic field measurements around
100 Hz, thus we set the minimum time scale separation value to $\delta=0.02$
s. In order to test the validity of the Markov property we evaluate both the
l.h.s. and the r.h.s. of the CK Equation (2) and we compare the resulting
transition probabilities. The l.h.s. of Equation (2) represents the empirical
transition probability $p_{E}$, whereas the r.h.s. is the CK transition
probability $p_{CK}$. The results of the CK test are reported in Figure 2 for
three different values of the scale separation: $\tau_{3}-\tau_{1}=0.02$ s,
$\tau_{3}-\tau_{1}=0.2$ s and $\tau_{3}-\tau_{1}=2$ s, being $\tau_{1}=0.02$
s. For sake of simplicity, the intermediate scale $\tau_{2}$ of the CK
integral is chosen as the half of the scale separation, although the results
are not affected by any particular choice of $\tau_{2}\in(\tau_{1},\tau_{3})$.
Figures 2a and 2d show an excellent agreement between $p_{E}$ and $p_{CK}$ at
0.02 s scale separation for $b_{1}$ and $b_{3}$, respectively. A similar
result can be observed at 0.2 s scale separation as reported in Figures 2b and
2e, thus pointing out that the magnetic field fluctuations $b_{i}(\tau)$ can
be successfully approximated as a Markov process in scale until the smaller
scales that are not affected by instrumental noise.

Figure 2: Comparison between $p_{E}$ (red curves) and $p_{CK}$ (blue curves)
at different timescales for minimum (upper panels) and maximum (lower panels)
variance directions. The timescale differences for the CK test are 0.02 s
(panels a and d), 0.2 s (panels b and e), and 1.0 s (panels c and f).

If the scale separation is pushed to 2 s, that corresponds to approach the low
end of the inertial range, the CK equation is still valid, but values of the
fluctuations $b_{i,\tau_{1}}$ observed at the smaller scale no longer depend
on the correspondent values $b_{i,\tau_{3}}$ observed at larger scales. This
result support the idea that IMF fluctuations observed in the sub-ion/kinetic
and inertial ranges are statistically independent, being perfectly in
agreement with previous findings in the near-Sun environment (Benella_2022, ).

The next step in the analysis is the evaluation of the first-, second- and
fourth-order KM coefficients in order to check whether or not the Pawula’s
theorem is fulfilled. In Figure 3 are reported the first-, second- and fourth-
order KM coefficients for both minimum and maximum variance components with
$\tau=0.06$ s and $\delta=0.02$. The fourth-order coefficient vanishes for
$B_{1}$ and $B_{3}$ in the case of Figure 3 (i.e., $\tau=0.06$ s) and for any
$\tau$ in the sub-ion/kinetic range (not shown). This confirms that the
evolution of the statistics of the process $b_{i}(\tau)$ as a function of
$\tau$ is governed by the FP equation, whose drift and diffusion coefficients
also depend on the scale.

One of the striking features of the sub-ion/kinetic range is the global-scale
invariance which is generally shown by rescaling the PDFs by introducing the
following transformation (kiyani2009global, ; alberti2019multifractal, ;
chhiber2021subproton, )

$\displaystyle b_{i,\tau}\rightarrow x_{i,\tau}=b_{i,\tau}/\sigma_{i,\tau}$
(9) $\displaystyle p(b_{i,\tau})\rightarrow
p(x_{i,\tau})=\sigma_{i,\tau}\,p(b_{i,\tau}).$ (10)

The existence of a master curve, i.e., a PDF invariant in shape, can be
interpreted in terms of stationary solution of the FP Equation (6) across sub-
ion/kinetic scales (Benella_2022, ). We aim to derive such master curve
starting from the KM coefficients evaluated for $b_{i,\tau}$. Since we are
dealing with finite-time KM coefficients instead of their “true” value (i.e.
taking the limit $\delta\to 0$), the relation between the KM coefficients
$D^{(k)}(b_{i},\tau)$ and $D^{(k)}(x_{i},\tau)$ is not straightforward. By
substituting the transformation (10) in Equation (8), we obtain the following
relations between the KM coefficients

Figure 3: First-, second- and fourth-order finite-scale KM coefficients for
minimum (left) and maximum (right) variance components of the IMF for
$\tau=0.06$ s and $\delta=0.02$ s.

$D^{(1)}_{\delta}(x_{i},\tau)=\frac{D^{(1)}_{\delta}(b_{i},\tau)}{\sigma}+\\\
\frac{1}{\delta}\biggl{(}\frac{1}{\sigma^{\prime}}-\frac{1}{\sigma}\biggr{)}\mathbb{E}[b_{i}(\tau-\delta)\rvert
b_{i}(\tau)],$ (11)
$D^{(2)}_{\delta}(x_{i},\tau)=\frac{D^{(2)}_{\delta}(b_{i},\tau)}{\sigma^{2}}+\frac{1}{2\delta}\biggl{(}\frac{1}{\sigma^{\prime
2}}-\frac{1}{\sigma^{2}}\biggr{)}\mathbb{E}[b_{i}^{2}(\tau-\delta)\rvert
b_{i}(\tau)]+\\\
-\frac{1}{\delta\sigma}\biggl{(}\frac{1}{\sigma^{\prime}}-\frac{1}{\sigma}\biggr{)}\mathbb{E}[b_{i}(\tau-\delta)b(\tau)\rvert
b_{i}(\tau)],$ (12)

where $\sigma$ and $\sigma^{\prime}$ are the standard deviations of the IMF
fluctuations at the scales $\tau$ and $\tau-\delta$, respectively. Starting
from the calculation of all the finite-scale KM coefficients at different time
scales within the sub-ion/kinetic range, Equations (11) and (12) enable us to
evaluate all the corresponding finite-scale KM coefficients of the rescaled
variables. By solving the FP equation for the stationary PDF, the general
solution reads

$p_{ST}(x_{i})=\mathcal{N}_{i}\exp\biggl{\\{}-\ln
D^{(2)}(x_{i})+\int_{-\infty}^{x_{i}}\frac{D^{(1)}(\xi)}{D^{(2)}(\xi)}d\xi\biggr{\\}}$
(13)

where $\mathcal{N}_{i}$ is the normalization factor.

Figure 4: Comparison between the empirical PDF of the rescaled variable
$x_{i}$ (circles) and the stationary solution of the FP equation (black lines)
for the minimum (a) and maximum (b) variance components. The time scales used
in the sub-ion/kinetic range range from 0.02 s to 0.2 s.

A fundamental remark of our reasoning is that by evaluating the functions
$D^{(1)}(x_{i},\tau)$ and $D^{(2)}(x_{i},\tau)$ through Equations (11) and
(12) at any $\tau$ and by substituting them in Equation (13), the obtained
stationary solution $p_{ST}(x_{i})$ approximates the empirical master curve
and does not depend on $\tau$. Figure 4ab shows the averaged function
$p_{ST}(x_{i})$ obtained through Equation (13) in the sub-ion/kinetic range
(black lines) along with the empirical rescaled PDFs (circles). The stationary
PDFs exhibit an excellent agreement with observations within
$x_{i}\in[-5\sigma_{x_{i}},5\sigma_{x_{i}}]$ for both minimum and maximum
variance components. We emphasize that the steady state solution has to be
interpreted as the analogous of the PDF-invariant shape observed in the sub-
ion/kinetic range, which stems from the fact that the FP Equation (6) describe
the evolution of PDFs as a function of the time scale $\tau$. It has been
shown that by parametrizing the KM coefficients as a linear ($D^{(1)}(x_{i})$)
and quadratic ($D^{(2)}(x_{i})$) functions of $x_{i}$, the master curve of
Figure 4 can be written as a Kappa distribution (Benella_2022, ). In this work
we have presented a more general derivation of the steady state solution in
the rescaled variable $x_{i}$ that is based on the evaluation of KM
coefficients on the variable $b_{i}$ at any $\tau$ the sub-ion/kinetic regime.

##  4 Discussion and conclusions

A universal framework of the solar wind processes originating magnetic field
fluctuations at sub-ion/kinetic scales is still lacking. Several mechanisms
can contribute in generating IMF fluctuations at these scales, e.g., wave-like
fluctuations, coherent magnetic structures, current patterns and magnetic
reconnection boldyrev2013toward ; lion2016coherent ; gary2009 ;
Schekochihin_2009 ; alexandrova2006 ; cerri2017reconnection . In this regard
the KM analysis of IMF fluctuations represents a novel approach to
characterize the stochastic nature of physical processes acting at these
scales. In this work we consider a fast solar wind stream in the near-Earth
environment and we have given evidence that the description of the PDF
evolution as a function of the time scale $\tau$ by means of Markov processes
is successful at sub-ion/kinetic scales. Such property has been also verified
for IMF fluctuations within the inertial range of magneto-fluid turbulence
(strumik2008statistical, ; strumik2008testing, ), then supporting the idea
that the energy flow from one scale to another has a local structure in both
inertial and sub-ion/kinetic regimes. Although there is a net energy transfer
between them, we provided evidence that a correspondence in terms of IMF
fluctuations between inertial and sub-ion/kinetic ranges does not seem to
emerge, confirming recent findings in the near-Sun environment (Benella_2022,
). This statistical independence between the observed fluctuations, together
with the transition between different dynamical regimes observed in the PSD,
supports the idea that the physical processes originating IMF fluctuations
have a different nature in the two regimes. We remark that these are
statistical results, therefore no direct information about the nature of the
specific physical processes can be inferred, even though our results provide
important constraints for models.

The main result reported in this work is represented by the clear Markovian
nature of the IMF fluctuations in the sub-ion/kinetic range in the near-Earth
solar wind. As pointed out in several recent works, in fact, many solar wind
characteristics, i.e. IMF fluctuations, exhibit a radial dependence in terms
of spectral features, intermittency, entropic character and so forth
chen2020evolution ; Alberti20 ; stumpo2021self . By performing the KM analysis
in the near-Earth IMF fluctuations we obtained similar results in comparison
with the near-Sun solar wind conditions. This allows us to assert that such
Markovian property of IMF fluctuations does not depend on the heliocentric
distance and thus might represent a universal property in the sub-ion/kinetic
domain.

The solar wind inertial range shows the typical anomalous scaling of fully
developed turbulence, with an intermittent character of the energy transfer,
whose strength depends on the heliocentric distance from the Sun Alberti20 .
In terms of scale dynamics of the IMF fluctuation statistics, the
intermittency is associated with a modification of the PDF across the inertial
range. This process affects both IMF-fluctuation amplitude as well as the
shape of their PDFs. Indeed, the higher probability of observing enhanced IMF
fluctuations towards smaller scales is reflected in the modification of the
Gaussian PDF at the integral scale into a leptokurtic function through the
energy cascade. In the framework of Markov processes, the PDF dynamics across
the inertial scales can be modeled through the general FP Equation, accounting
for the variation of the shape of the PDF as a function of $\tau$. As a
consequence, the absence of modification of the initial PDF across different
scales can be due to the lack of intermittency in the physical processes
accounting for the energy transfer. In this work we confirmed that the global
scaling observed at sub-ion/kinetic scales can be interpreted by means of the
stationary FP Equation (i.e., $\partial p/\partial\tau\to 0$) since the PDFs
of IMF fluctuations exhibit a well-defined shape-invariant form in this
domain. From the physical point of view our results can be resembled in the
following scenario:

  * •

as for the inertial range, also at sub-ion/kinetic scales the energy transfer
is local in scale, i.e., it satisfies the Markov property, providing a
complete description of the IMF fluctuations in terms of Langevin dynamics:
the energy transfer turns out to be a stochastic process;

  * •

the energy transfer mechanisms in the inertial range and in the sub-
ion/kinetic domains originate IMF fluctuations exhibiting a statistical
independence between the two regimes, thus they may have a different nature;

  * •

the intermittency of IMF fluctuations observed in the inertial range is lost
at sub-ion/kinetic scales, where a global scale invariance is observed.

A joint analysis involving both high-resolution magnetic and velocity field is
desirable and will provide a more comprehensive picture of the statistics of
the physical processes acting at sub-ion/kinetic scales.

Acknowledgments

This work is funded by the Italian MIUR-PRIN grant 2017APKP7T on
“Circumterrestrial Environment: Impact of Sun-Earth Interaction”. The authors
acknowledge the Cluster FGM and STAFF P.I.s and teams and the ESA-Cluster
Archive for making available the data used in this work. The FGM and STAFF
data were combined using the IRFU-Matlab analysis package available at
https://github.com/irfu/irfu-matlab. M.S. acknowledges the PhD course in
“Astronomy, Astrophysics and Space Science” of the University of Rome
“Sapienza”, University of Rome “Tor Vergata” and Italian National Institute
for Astrophysics (INAF), Italy. E.Y. is supported by SNSA grants 86/20 and
145/18.

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