# Sign singularity of the local energy transfer in space plasma turbulence

Luca Sorriso-Valvo  Nanotec/CNR, U.O.S. di Cosenza, Ponte P. Bucci, cubo 31C,
87036 Rende, Italy  Departamento de Física, Escuela Politécnica Nacional,
Quito, Ecuador  Gaetano De Vita  Nanotec/CNR, U.O.S. di Cosenza, Ponte P.
Bucci, cubo 31C, 87036 Rende, Italy  Federico Fraternale  Dipartimento di
Scienza Applicata e Tecnologia, Politecnico di Torino, Torino, Italy
Alexandre Gurchumelia  M. Nodia Institute of Geophysics, Iv. Javakhishvili
Tbilisi State University, Tbilisi, Georgia  Silvia Perri  Dipartimento di
Fisica, Università della Calabria, Rende, Italy  Giuseppina Nigro
Dipartimento di Fisica, Università della Calabria, Rende, Italy  Filomena
Catapano  Serco Italia for ESA-ESRIN, Frascati, Italy  Alessandro Retinò  LPP-
CNRS/Ecole Polytechnique/Sorbonne Université, Paris, France  Christopher H. K.
Chen  School of Physics and Astronomy, Queen Mary University of London, UK
Emiliya Yordanova  Swedish Institute of Space Physics, Uppsala, Sweden  Oreste
Pezzi  Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila, Italy
INFN/Laboratori Nazionali del Gran Sasso, Assergi, Italy  Khatuna Chargazia
M. Nodia Institute of Geophysics, Iv. Javakhishvili Tbilisi State University,
Tbilisi, Georgia  I. Vekua Institute of Applied Mathematics, Iv. Javakhishvili
Tbilisi State University, Tbilisi, Georgia  Oleg Kharshiladze  M. Nodia
Institute of Geophysics, Iv. Javakhishvili Tbilisi State University, Tbilisi,
Georgia  Diana Kvaratskhelia  M. Nodia Institute of Geophysics, Iv.
Javakhishvili Tbilisi State University, Tbilisi, Georgia  Sokhumi State
University, Tbilisi, Georgia  Christian L. Vásconez  Departamento de Física,
Escuela Politécnica Nacional, Quito, Ecuador  Raffaele Marino  Laboratoire de
Mécanique des Fluides et d’Acoustique, CNRS, École Centrale de Lyon,
Université Claude Bernard Lyon 1, INSA de Lyon, F-69134 Écully, France
Olivier Le Contel  LPP-CNRS/Ecole Polytechnique/Sorbonne Université, Paris,
France  Barbara Giles  NASA, Goddard Space Flight Center, Greenbelt MD 20771,
USA  Thomas E. Moore  NASA, Goddard Space Flight Center, Greenbelt MD 20771,
USA  Roy B. Torbert  Space Science Center, University of New Hampshire,
Durham, New Hampshire, USA  James L. Burch  Southwest Research Institute, San
Antonio, Texas, USA

###### Abstract

In weakly collisional space plasmas, the turbulent cascade provides most of
the energy that is dissipated at small scales by various kinetic processes.
Understanding the characteristics of such dissipative mechanisms requires the
accurate knowledge of the fluctuations that make energy available for
conversion at small scales, as different dissipation processes are triggered
by fluctuations of a different nature. The scaling properties of different
energy channels are estimated here using a proxy of the local energy transfer,
based on the third-order moment scaling law for magnetohydrodynamic
turbulence. In particular, the sign-singularity analysis was used to explore
the scaling properties of the alternating positive-negative energy fluxes,
thus providing information on the structure and topology of such fluxes for
each of the different type of fluctuations. The results show the highly
complex geometrical nature of the flux, and that the local contributions
associated with energy and cross-helicity nonlinear transfer have similar
scaling properties. Consequently, the fractal properties of current and
vorticity structures are similar to those of the Alfvénic fluctuations.

magnetosphere, turbulence, dissipation

###### pacs:

94.05.-a, 94.05.Lk, 95.30.Qd

##  I Introduction

The dynamics of space plasmas is characterized by a broad variety of complex
processes that include turbulence, instabilities, and several mechanisms of
particle-radiation interaction. Such processes are intrinsically connected
across multiple scales. For example, the energy associated with large-scale
structures and instabilities is transported towards smaller and smaller scales
though a turbulent cascade due to the nonlinear interactions among magnetic
and velocity fluctuations, throughout the so-called inertial range that may
span one to more than three decades in scales Frisch (1995); Marsch & Tu
(1997); Bruno & Carbone (2013). When the energy reaches scales of the order of
or smaller than the typical ion and electron scales (e.g. the proton Larmor
radius or inertial length), a different turbulent cascade occurs Macek et al.
(2018); Chen and Boldyrev (2017). At those scales, weakly collisional plasma
kinetic processes arise, such as nonlinear damping of waves, kinetic
instabilities, particle collisions, and magnetic reconnection, that convert
the energy stored in the field fluctuations into particle energization and
acceleration, and plasma heating Salem et al. (2012); Chang, Gary and Wang
(2015); Chen (2016); Pezzi, Valentini and Veltri (2016); Pezzi (2017); Chen et
al. (2019).

Past theoretical, experimental and numerical attempts to describe these
processes have focused mostly on simplified, idealized conditions. However, in
recent years there is an increasing interest in their cross-scale, interwoven
nature. Multi-spacecraft and high resolution measurements in the solar wind
and in the terrestrial magnetosphere Escoubet et al. (2001); Burch et al.
(2016a) have provided evidence of such interconnection Greco et al. (2016).
The increasing performance of numerical simulations has also allowed processes
on several scales to be examined, and therefore to highlight their
relationship Karimabadi et al. (2013); Valentini et al. (2014); Franci et al.
(2015); Servidio et al. (2015); Valentini et al. (2016); Cerri, Servidio and
Califano (2017); Groselj et al. (2017); Franci et al. (2018); Pezzi et al.
(2018); Perrone et al. (2018). Theoretical efforts are also being carried out
in order to highlight the specific processes governing the energy exchange
between ranges associated to different regimes (Schekochihin et al., 2016;
Servidio et al., 2017; Sorriso-Valvo et al., 2018a). In this framework, the
local, fine-scale details of the turbulent energy cascade acquire new
importance, as the specific characteristics of the fluctuations carrying
energy to the kinetic scales can be associated with different plasma processes
(Klein et al., 2016; Sorriso-Valvo et al., 2019; Chen et al., 2019).

Recent analysis has revealed that the temperature and energized particles are
enhanced in the proximity of current sheets Osman et al. (2012); Servidio et
al. (2012); Vásconez et al. (2015); Pucci et al. (2016); Valentini et al.
(2017) or of locations of concentration of turbulent energy Sorriso-Valvo et
al. (2018a, b). The local, fine details of the energy transfer process in the
turbulent cascade may therefore play a fundamental role in the activation of
those plasma kinetic processes that are believed to be responsible for energy
conversion, usually (and loosely) referred to as dissipation. In numerical
simulations specific techniques, mostly based on Fourier-space filtering, have
been developed to achieve a detailed description of the energy transfer
Camporeale et al. (2018); Yang et al. (2019); Kuzzay et al. (2019). However,
the limitations arising from the one-dimensional nature of spacecraft sampling
require the introduction of approximated quantities. A simple example is
provided by the normalized magnitude of the small-scale magnetic field
fluctuations, basically locating current sheets and similar magnetic
structures. Techniques known as local intermittency measure (LIM) Farge (1992)
and partial variance of increments (PVI) Greco et al. (2008) were extensively
used in the last decades. A more informative proxy, called local energy
transfer (LET), is based on the third-order scaling law for turbulent plasmas
Politano & Pouquet (1998), and carries information about the nature of the
fluctuations transporting the energy to small scales Sorriso-Valvo et al.
(2019). For example, the use of this proxy allowed the identification of
specific ion features, such as beams, where the alignment between small-scale
magnetic field and velocity fluctuations was dominating. This suggested
nonlinear resonance between Alfvénic fluctuations and particles as a possible
mechanism for the generation of those beams Sorriso-Valvo et al. (2019).

In this article, the topological properties of the energy flow channels are
examined. Measurements from the solar wind and from different regions of the
terrestrial magnetosphere are studied by means of sign-singularity analysis.
The results show the presence of interwoven positive-negative energy flux,
allowing estimation of the typical fractal dimension of the structures, and
eventually the role of their different components, in the turbulent cascade
and, therefore, on feeding small-scale dissipative processes. Section II
describes the proxy used in this work and the cancellation analysis technique.
In Section III we describe the data used. Section IV provides a description of
the results and the comparison between different data sets. Finally, the
results are briefly discussed in Section Discussion.

##  II Methods

###  II.1 A proxy of the local energy transfer in turbulence

The fluctuations observed in magnetohydrodynamic plasma turbulence have been
shown to follow the Politano-Pouquet law (Politano & Pouquet, 1998). This
predicts the linear scaling of the mixed third-order moment of the fields
fluctuations on the scale, when homogeneity, scale separation, isotropy, and
time-stationarity are met. Using the Taylor hypothesis (Peyret & Taylor, 1986;
Perri et al., 2017) $\mathbf{r}=t\langle\mathbf{v}\rangle$ (necessary to
transform space ($\mathbf{r}$) and time ($t$) arguments via the bulk speed
$\langle\mathbf{v}\rangle$), the Politano-Pouquet law can be written as

$Y({\Delta t})=\langle\Delta
v_{l}(|\Delta\mathbf{v}|^{2}+|\Delta\mathbf{b}|^{2})-2\Delta
b_{l}(\Delta\mathbf{v}\cdot\Delta\mathbf{b})\rangle=-\frac{4}{3}\langle\varepsilon\rangle\Delta
t\langle v\rangle\;.$ (1)

The mixed third-order moment $Y({\Delta t})$ is computed using the increments
$\Delta\psi(t,\Delta t)=\psi(t+\Delta t)-\psi(t)$ of a field $\psi$ (either
the plasma velocity $\mathbf{v}$ or the magnetic field
$\mathbf{b}=\mathbf{B}/\sqrt{4\pi\rho}$ given in velocity units through the
mass density $\rho$) across a temporal scale $\Delta t$, the subscript $l$
indicating the longitudinal component, i.e. parallel to the bulk speed. The
total energy flux given in equation (1) is proportional to the mean energy
transfer rate $\langle\varepsilon\rangle$. The Politano-Pouquet law describes
the scaling of the small imbalance between positive and negative energy flux
in the turbulent cascade, and is associated with the scale-dependent intrinsic
asymmetry (skewness) of the turbulent fluctuations Frisch (1995); Politano &
Pouquet (1998). The linear scaling (1) was robustly observed in numerical
simulations (Sorriso-Valvo et al., 2002; Andrés et al., 2018; Ferrand et al.,
2019), in the solar wind plasma (MacBride et al., 2005; Sorriso-Valvo et al.,
2007; Marino et al., 2008; Coburn et al., 2012; Banerjee et al., 2016), and in
the terrestrial magnetosheath (Hadid et al., 2018; Bandyopadhyay et al.,
2018a, b). In order to attempt a description of the local energy flux from
space data time series, the law (1) can be revisited without computing the
average, thus giving a time series of the heuristic proxy of the local energy
transfer rates (LET) at a given scale ${\Delta t}$, which can be estimated by
computing the quantity:

$\varepsilon^{\pm}(t,{\Delta t})=-\frac{3}{4}\frac{\Delta
v_{l}(|\Delta\mathbf{v}|^{2}+|\Delta\mathbf{b}|^{2})-2\Delta
b_{l}(\Delta\mathbf{v}\cdot\Delta\mathbf{b})}{{\Delta t}\langle v\rangle}\,.$
(2)

This procedure neglects several contributions to the scaling, which in (1) are
suppressed by averaging over a large sample, and therefore provides only a
rough approximation of the actual local energy transfer rate Hellinger et al.
(2018); Kuzzay et al. (2019). However, because of the intrinsic difficulty in
estimating the neglected terms from one-dimensional data, this proxy can be
used as a first degree approximation in space plasmas time series. The LET was
previously used to determine heating regions in the interplanetary plasma
(Sorriso-Valvo et al., 2018a) and on kinetic numerical simulations (Sorriso-
Valvo et al., 2018b; Yang et al., 2019).

The LET is composed of two additive terms, one associated with the magnetic
and kinetic energy advected by the velocity fluctuations,
$\varepsilon_{e}=-3/(4\Delta t\langle v\rangle)[\Delta
v_{l}(\Delta\mathbf{v}^{2}+\Delta\mathbf{b}^{2})]$, and the other with the
cross-helicity coupled to the longitudinal magnetic fluctuations,
$\varepsilon_{c}=-3/(4\Delta t\langle v\rangle)[-2\Delta
b_{l}(\Delta\mathbf{v}\cdot\Delta\mathbf{b})]$ (Sorriso-Valvo et al., 2018a).
Such separation has been used to identify regions dominated by current and
vorticity structures from regions dominated by coupled, Alfvénic fluctuations
in the terrestrial magnetospheric boundary layer, revealing the presence of
ion beams mostly associated with the small-scale Alfvénic fluctuations, and
thus indicating a possible mechanism for the transfer of the turbulent energy
to the particles (Sorriso-Valvo et al., 2019). Since the LET, as well as its
two separated components, are signed quantities, it may be interesting to
explore the scaling properties of the mixing of the positive and negative
parts of the turbulent cascade. These may be related to the direction of the
energy flow, although this interpretation is not supported by theoretical
evidence. Moreover, unlike in the averaged Politano-Pouquet law, decoupling
the sign dependence on scale and position is not trivial for the local proxy.
Therefore, caution should be used in evaluating the physical meaning of the
sign. However, it could still be associated to injection or removal of energy
from specific locations and scales. It has been shown both in MHD Debliquy et
al. (2005) and in hydrodynamic flows Shaoo and Biferale (2015) that a
selective filter of the triads carrying the energy throughout the inertial
range, as well as the absence of resonant triads in the anisotropic case in
the presence of rotation and stratification Marino et al. (2013), may lead to
the modulation of an inverse cascade in fully developed three-dimensional
turbulence. It would be thus interesting to investigate the nature of the sign
of the local energy dissipation obtained with the proxy proposed here also by
the implementation of shell models Verma et al. (1996); Dar et al. (2001).

Preliminary comparison between the proxy and more comprehensive estimates of
the local energy transfer rate, performed using three-dimensional MHD
numerical simulations Kuzzay et al. (2019), suggests good qualitative
agreement (not shown) in terms of location of the larger transfer regions,
although there are some discrepancies in the magnitude and fluctuation of the
signed transfers that may relate to the approximated and unfiltered nature of
the LET. For the purposes of this study, the proxy does not necessarily need
to fully capture the turbulent energy flux, as it is rather related to the
specific features of the plasma and field fluctuations that contribute to the
actual energy flux.

The complexity of the energy flow across scales might carry information about
the topology of the small-scale structures, and also specifically for their
energy or cross-helicity contributions. This information can be useful to
determine which dissipative processes are selectively activated by the
turbulent cascade. In this work, we aim at providing such information, that
will be obtained by means of the cancellation analysis, which is briefly
described in the following section.

###  II.2 Sign singularity and cancellation analysis

The properties of chaotic flows can be described through the singularity
analysis of the field (Ott et al., 1992). In particular, if a given field
changes sign on arbitrarily small scale, its measure is called sign singular
(Ott et al., 1992). The quantitative description of this singularity is
important for the description of the topological properties (e.g. fractal
dimension, filling factor…) of sign-defined (smooth) coherent structures, such
as the ones emerging in intermittent, turbulent flows. A standard technique to
estimate sign singularity is provided by the cancellation analysis, previously
used to describe the scaling properties of MHD, Hall-MHD and Vlasov-Maxwell
turbulence in numerical simulations (Sorriso-Valvo et al., 2002; Graham et
al., 2005; Martin et al., 2013; De Vita et al., 2014; Gingell et al., 2017)
and in the current helicity in solar photospheric active regions (Abramenko et
al., 1998; Sorriso-Valvo et al., 2004, 2015; De Vita et al., 2015).

Given a scalar field $f({\bf r})$ with zero mean, defined on a $d$-dimensional
domain $Q(L)$ of size $L$, its signed measure can be defined as the normalized
field integrated over scale dependent subsets $Q(l)\subset Q(L)$ of size $l$,

$\mu(l)=\frac{\int_{Q(l)}\;\mathrm{d}{\bf r}\;f({\bf
r})}{\int_{Q(L)}\;\mathrm{d}{\bf r}\;|f({\bf r})|}\,.$

A coarse-graining of the domain provides an estimate of the sign-singularity
of the measure by means of the scaling exponent $\kappa$ (also called
cancellation exponent) of the cancellation function, which is in turn defined
as $\chi(l)=\sum_{Q_{i}(l)}|\mu_{i}(l)|\sim l^{-\kappa}$, the sum being
intended over all disjoint subsets $Q_{i}(l)$ fully covering the domain
$Q(L)$. In a chaotic field, positive and negative fluctuations cancel each
other if the integral is performed over large subsets, resulting in a small
signed measure at large scales. However, if the integration subset has the
typical size of the smooth structures, cancellations are reduced and the
signed measure is relatively larger. The scaling law of the cancellation
function, as described by the cancellation exponent, can thus provide
information on the field cancellations across the scales. Some specific values
of the cancellation exponent can help to interpret the results. If the field
is smooth, then the cancellation function does not depend on the scale, and
$\kappa=0$. If the field is homogeneous with random discontinuities, then
cancellations are enhanced and $\kappa=d/2$. Values in between these two
examples indicate the coexistence of random fluctuations and smooth
structures, whose fractal dimension $D$ is thus given by $\kappa=(d-D)/2$
(Sorriso-Valvo et al., 2002). The fractal dimension $D$ provides information
about the space filling and complexity of the structures carrying the energy
to small scales, and might be related to the efficiency of the transport
mechanism. In this study we will make use of $\kappa$ and $D$ as parameters to
describe the topological properties of the different turbulent energy
channels, and compare the results for the interplanetary space and the
magnetosphere.

##  III Data

In order to study the cancellation properties of the local energy transfer
rate proxy LET, and of its two components, we have selected two magnetospheric
plasma intervals measured by the Magnetospheric Multiscale mission (MMS) Burch
et al. (2016a), which provides data at high cadence, and one longer interval
of fast solar wind measured by the Wind spacecraft Acuña et al. (1995).

The first sample, labeled as MMS-KH, was recorded on September 8, 2015 between
10:07:04 and 11:25:34 UTC, while MMS was in the dusk-side magnetopause, moving
across a portion of plasma dominated by the Kelvin-Helmholtz instability (KH)
formed at the boundary between the magnetosheath and the magnetosphere. The
interval was extensively studied in the past, showing the presence of strong
turbulence and intermittency Stawarz et al. (2016). The MMS spacecraft
performed multiple crossings of the KH boundary, resulting in the alternate
sampling of plasma from the magnetosheath and from the magnetospheric boundary
layer. The boundary crossings between the two regions are clearly highlighted
by sharp transitions of the plasma parameters, so that it is easy to separate
them. In this work, we have accurately selected 53 short intervals (ranging
one to ten minutes) that are purely immersed in the magnetospheric boundary
layer (based on plasma temperature and density), rejecting magnetosheath and
transition regions. This allows some degree of homogeneity of the sample,
necessary for statistical analysis.

The second magnetospheric interval, named MMS-MS, was selected in the
turbulent magnetosheath region under quasi-parallel bow shock geometry on
November 30, 2015, between 00:21 and 00:26 UTC. This 5-minute interval is
characterized by intense fluctuations in all plasma and field parameters and
by the presence of small scale magnetic structures. Some of these have been
studied in detail, and various kinetic processes, such as local electron
acceleration and magnetic reconnection at thin current sheets have been
observed Eriksson et al. (2016); Yordanova et al. (2016); Vörös et al. (2017).

For both MMS intervals, the magnetic field data with sampling frequency $1$
kHz used here are a merged product Fischer et al. (2016) from the burst mode
flux gate (FGM) Russell et al. (2016) and search coil (SCM) Le Contel et al.
(2016) instruments on MMS. The ion moments come from the fast plasma
instrument (FPI) Pollock et al. (2016) at a sampling rate of $150$ ms.

Finally, in order to compare the magnetospheric results with the solar wind,
we have also studied the cancellation properties of the proxy LET using one
sample of fast solar wind measured by the Wind spacecraft Acuña et al. (1995),
labeled as WIND. The data interval consists of 6 days when Wind was in a fast
stream during days 14 to 19 of 2008, and is the same interval as studied in
Wicks et al. (2013). The data from the magnetic field instrument MFI (Lepping
et al., 1995) and plasma instrument 3DP (Lin et al., 1995) at $3$ s cadence
were used, with the magnetic field converted to Alfvén units using the kinetic
normalization described in Chen et al. (2013). The average conditions during
the interval were a solar wind speed of 660 km/s, magnetic field strength B =
4.4 nT, density 2.4 cm-3 and proton beta $\beta_{p}=1.2$, typical for the fast
solar wind.

The three intervals are characterized by variable levels of magnetic
fluctuations. As shown in Figure 1, all three intervals present a reasonably
well defined power-law spectral scaling range. For the WIND and MMS-KH
intervals, the spectra are close to the Kolmogorov prediction, with scaling
exponents $\sim-5/3$ compatible with the standard values for MHD turbulence.
The MMS-KH data shows slightly shallower spectra, but still in the standard
range of observation of turbulent space plasmas. Note that the power spectra
in these data were obtained using the compressed sensing technique described
in Fraternale et al. (2018). The magnetosheath interval, MMS-MS, has less
defined power-law scaling, possibly because of its short duration, and the
scaling exponent is $\sim-2$ (see the fitted exponents inside each panel
frame), suggesting the presence of uncorrelated structures. This is typical of
the highly fluctuating magnetosheath magnetic field, and indicates a
relatively less developed turbulence. In the magnetosheath flanks, Kolmogorov-
type power spectra can be observed in the MHD range Alexandrova et al. (2008);
Huang et al. (2017). However, in the region closer to the subsolar point,
where MMS orbit lies during this particular event, the plasma is highly
compressed and closely confined between the bow shock and the magnetopause.
The solar wind turbulence, once modified and shuffled by the bow shock
crossing, does not have enough space and time to reach a fully developed
state, because of the close proximity of the two large boundaries. This
results in the observed steeper spectral exponents. Note that in the present
sample the typical ion frequencies are of the order of $1.3$ Hz, which exclude
the possibility that the observed scaling range is in the kinetic regime Macek
et al. (2018).

Figure 1: Magnetic components spectral power density for the three samples.
The trace is shown in black, scales by a factor of ten for clarity. For MMS-KH
(left panel), both the compressed sensing spectra (thin curves) and the
spectra averaged over the sub-intervals (thicker curves) are shown. The
standard deviation of all 53 sub-intervals is shown in gray for the trace. The
blue-dashed vertical line corresponds to the large-scale KH frequency
$f_{\mathrm{KH}}=0.0146$ Hz. For MMS-MS and WIND (central and right panels),
the unsmoothed trace is shown with gray points. Fitted power-law exponents in
the MHD-inertial frequency range indicated by grey vertical lines are given.
Reference power laws with $-5/3$ exponent are also shown.

Figure 2: Scale-dependent kurtosis of the magnetic field components, for the
three samples. The fitted power-law exponents are shown for the MHD inertial
regime (color coded), delimited by gray vertical lines. Power laws with
exponent $-1/3$ are indicated as a reference (gray).

Figure 3: The Politano-Pouquet law (1) for the three samples, indicated as
$\langle\varepsilon\rangle$ in the legend (black), and in terms of its
averaged components $\langle\varepsilon_{e}\rangle$ (red) and
$\langle\varepsilon_{c}\rangle$ (blue). Negative terms are represented as full
lines, while the inverted positive terms as dashed lines. The thin grey lines
represent linear scaling law, and are shown for reference.

The formation of small-scale structures, typical of intermittency, is
evidenced by the (roughly) power-law increase towards the small scales of the
normalized fourth order moment (kurtosis) of the magnetic fluctuations,
$K_{i}(\Delta t)=\langle\Delta B_{i}^{4}\rangle/\langle\Delta
B_{i}^{2}\rangle^{2}$ (Figure 2), the subscript $i$ indicating the component
$x$, $y$ or $z$. Note that the Gaussian value $K=3$ is observed for scales
larger than the estimated inertial range (right grey vertical line). The
Power-law decrease with the scale is a direct consequence of the structure
function scaling in turbulent fields. The fitted scaling exponents are
indicated in each panel, and are proportional to the degree of intermittency
of the system Carbone and Sorriso-Valvo (2012). For the solar wind data, where
the turbulence is more developed, the power-law behaviour is more evident.
Both the exponents and the small-scale magnitude of the kurtosis are in
agreement with typical values for fast solar wind Bruno & Carbone (2013). A
shorter, less defined power-law scaling range, with slightly smaller scaling
exponents, is observed in MMS-KH and MMS-MS data, suggesting a less developed
intermittency in the younger turbulence of the shocked plasma. Similar results
(not shown) were obtained through the standard analysis of the anomalous
scaling of the structure functions Frisch (1995), fitted to a $p$-model
Meneveau and Sreenivasan (2012), in the extended self-similarity approach
Benzi et al. (1993). After averaging over the three components (no major
differences were observed), the magnetic field intermittency parameters $p$
are $0.67$, $0.79$, and $0.82$ for the MMS-KH, MMS-MS, and WIND samples
respectively ($p$ lies in the interval $[0.5,1]$, with $p=0.5$ indicating
absence of intermittency). These results show the strongly intermittent nature
of the WIND sample, the slightly less intermittent MMS-SH sample, and the
weakly intermittent nature of the MMS-KH sample.

Finally, the Politano-Pouquet law (1) can be estimated in the samples under
study, both in terms of total energy transfer $\langle\varepsilon\rangle$, and
in terms of the averaged components $\langle\varepsilon_{e}\rangle$ and
$\langle\varepsilon_{c}\rangle$. The resulting scaling functions are shown in
Figure 3 for the three intervals. None of the observed cases display a clear
linear scaling. This might be due to the violation of the several requirements
necessary for the Politano-Pouquet law to hold (e.g. incompressibility,
isotropy, stationarity, large Reynolds number), to the presence of large-scale
features advected by the flowing plasma, or simply to the lack of statistical
convergence of the third-order moment, due to intrinsic finite-size limitation
of space data. The challenging observation of the linear law in solar wind was
already noticed using Ulysses data Sorriso-Valvo et al. (2007); Marino et al.
(2012).

In the MMS-KH data a power-law scaling slightly different from the expected
linear relation is suggested. In WIND, there is evidence of linear scaling of
the two components separately, while their combination does not display the
predicted linear dependence. In the MMS-MS sample, the third-order moment and
its components experience multiple sign inversions, possibly due to the finite
size sample and to the expected poorly developed turbulence, evidenced by the
steeper magnetic spectral exponent, as discussed above.

The intervals used in this work appear thus characterized, to different
degrees, by the presence of an inertial range of turbulent, intermittent
fluctuations, with a roughly defined energy cascade leading to the formation
of small-scale structures.

##  IV Results

From the MMS and WIND measurements described above, we have computed the LET,
examples of which are shown in Figure 4. In this work we will use the LET at
scale of $1.2$ s, which is still inside and near the bottom of the MHD
inertial range, where the Politano-Pouquet law is valid Sorriso-Valvo et al.
(2019). The proxy has the typical behavior of intermittent dissipation in
turbulence Frisch (1995), with the presence of intense bursts of energy flux
alternating with quieter regions. The cancellation analysis described in
Section II was then performed on the signed fields $\varepsilon$,
$\varepsilon_{e}$ and $\varepsilon_{c}$, as obtained from the different data
sets considered for this study. A range of time-scale separations within the
inertial range was considered, so that we have estimates of the LET for
different scales $\Delta t$ within the turbulent cascade.

Top panels of Figure 5 shows three examples of scaling of the cancellation
function $\chi(l)$ for the LET proxy $\varepsilon$, computed using the field
fluctuations at a scale $\Delta t$ near the bottom of the inertial range, as
indicated in each figure. Each example refers to one of the three data sets
studied in this paper. Power-law scaling can be easily identified in a region
roughly corresponding to the respective inertial range of MHD turbulence (see
Figure 1 for comparison). In the WIND data, a possible secondary power-law
scaling is observed in the large-scale range $\Delta t\gtrsim 10$ m, where
spectra usually decay as $1/f$ (see in Figure 1 the large-scale spectral break
at $f\simeq 0.002$ Hz) Bruno & Carbone (2013). On the other hand, the higher
resolution of MMS data allows to highlight the presence of scaling in the ion
range of scales (i.e. for $\Delta t\lesssim 1$ s, compatible with the spectral
break visible near 1 Hz in Figure 1), where a different type of cascade may
take place Alexandrova et al. (2008); Stawarz et al. (2016); Macek et al.
(2018). However, the scaling in this range should be studied in the framework
of ion plasma physics, for example by including the Hall-MHD corrections to
the Politano-Pouquet law Hellinger et al. (2018); Ferrand et al. (2019). This
is left for future study. The cancellation functions have been fitted to power
laws in the inertial range for all samples, and additionally in the $1/f$
scaling range for the solar wind data, providing the cancellation exponents
$\kappa$, and thus the corresponding fractal dimensions $D=d-2\kappa$ (in this
case $d=1$, so that the values of $D=1$ would indicate smooth, space-filling
structures). Their values are indicated in the figure for some selected
examples. A similar behaviour was robustly observed for all samples and all
LET components, and at all scales $\Delta t$ within the inertial range, so
that it is possible to compare the cancellation properties of the LET in the
different cases.

Figure 4: The local proxy $\varepsilon$ as a function of time, estimated at
the scale $\Delta t=1.2$s for the two MMS intervals, and at $\Delta t=6$s for
WIND. Note that in the MMS-KH sample, the displayed signal results from the
aggregation of the 53 separated sub-intervals, and has been displayed
continuously in order to highlight its global properties.

Figure 5: Top row: the cancellation function $\chi(l)$ of the LET
$\varepsilon$ for the three data sets. Power-law fits are indicated in the
range of scale roughly corresponding to the spectral inertial range. Bottom
row: the scale-dependent cancellation exponent $\kappa$, as computed from the
fit in the MHD range of the cancellation function $\chi(l)$ of the three LET
proxies $\varepsilon$ (full lines), $\varepsilon_{e}$ (dashed lines) and
$\varepsilon_{c}$ (dash-doted lines), shown as red lines for the three data
sets. In the right panel (WIND), the blue lines (same styles as above) refer
to the proxies estimated in the $1/f$ range.

In the MMS-KH interval (left panel), the cancellation exponents near the end
of the inertial range ($\Delta t=1.2$ s) is $\kappa\sim 1/3$ for the three
variables, a value indicating high complexity. This corresponds to fractal
dimension of the order of $D\sim 0.33$, which is indicative of highly
fragmented structures. In the magnetosheath sample MMS-MS (central panel), at
the same scale the exponent is closer to $\kappa\sim 1/2$, which is usually
representative of random sign alternation, or absence of smooth, persistent
structures of that scale. This is in agreement with the observed steep
spectrum, indicative of the presence of weakly correlated discontinuities (or
structures), and with the large kurtosis of this sample, which accounts for
the broad presence of such structures. In the fast solar wind sample measured
by Wind, the scaling exponent of the total energy transfer proxy near the
bottom of the inertial range ($\Delta t=6$s) is $\kappa\sim 1/4$, which
corresponds to the presence of structures of fractal dimension $D\sim 0.5$.
This is in agreement with the typical observation of disrupted current sheets
of solar wind intermittent turbulence Martin et al. (2013), and confirms the
fact that the turbulence is well developed in this fast wind stream, with
strong intermittency. For the same interval, in the $1/f$ range of scales, a
different fit of the cancellation function provides $\kappa\sim 1/2$, in
excellent agreement with the uncorrelated nature of the Alfvénic fluctuations
observed at such scales Bruno & Carbone (2013).

Further information can be gained by observing the scale dependence of the
cancellation exponent (or the corresponding fractal dimension). This can be
obtained using the LET proxy estimated at different scales, using increments
of the fields on variable scales $\Delta t$. Thus, for each scale, the LET and
its components provide scale-dependent, local estimates of the turbulent
energy flow. Results of cancellation analysis are collected in the bottom
panels of Figure 5, where the cancellation exponent $\kappa$ is shown for the
three samples, and for the three variables. For all samples, at scales larger
than the correlation scale (roughly $10$ s for both MMS samples Stawarz et al.
(2016); Sorriso-Valvo et al. (2019) and about $30$ s for WIND, as evident from
the spectrum and from the kurtosis) the cancellation exponent is
$\kappa\lesssim 0.1$ (or $D\gtrsim 0.8$) for all fields, as expected for
smooth, space-filling fluctuations. As the scale decreases, all samples
display an increase of complexity, in a scale range roughly corresponding to
the inertial range, where the intermittent structures are generated. Finally,
a plateau or saturation seems to take place at or near the ion-scale spectral
break. This could indicate that the intermittent structures have reached their
stable geometry. However, this effect could also be due to the MHD nature of
the LET proxies, which might be unable to properly capture further
fragmentation of the fluctuations. The study of the ion range with the
appropriate variable is left for future work.

In the MMS-KH interval (left panel), the increase of complexity towards small
scales is smooth and power-law like, and extends to the whole inertial range.
The nonlinear cascade and the complex entanglement of positive and negative
energy flux (proxies of the direct and inverse cascade, respectively) is thus
beautifully captured by the LET in this sample. In the inertial range, all
three variables (different lines) have similar exponents. The cross-helicity
component $\varepsilon_{c}$ seems to provide slightly larger exponents than
the energy component $\varepsilon_{e}$, suggesting that the energy transfer
associated with current and vorticity structures occurs in a slightly smaller
fraction of the volume (smaller fractal dimension).

As for the KH interval, the WIND cancellation exponents obtained in the MHD
range present similar exponents for the three MHD variables throughout the
whole range of scales, indicating that the alternation of positive and
negative energy flow is similar for the three proxies. This could be an
indication of well developed turbulence, where a sufficient equilibrium
between the competing terms in the cascade has been reached. The growth of the
complexity roughly follows a power-law scaling, which confirms the excellent
scaling properties of this sample, and that the LET proxies capture the sign
complexity of the energy cascade.

From the bottom panels of Figure 5, it is evident that the overall behavior
described above is roughly coherent for the WIND (in the MHD range) and MMS-KH
samples. This suggest the universality of the turbulent cascade mechanism, at
least with respect to the sign-singularity properties, or, equivalently, to
the fractal properties of the structures responsible for the energy transfer.
The similarity is also corresponding to the presence of well-developed
Kolmogorov spectra for both samples (see Figure 1), and to the power-law
scaling of the kurtosis describing intermittency. In particular, the small-
scale limiting values of $\kappa$ are larger for the MMS-KH sample than for
WIND, in agreement with the more developed intermittency highlighted by the
kurtosis and structure function analysis.

In the magnetosheath data, the increase of complexity of the positive-negative
alternation for the total and structure-related proxies is sharper, less
defined, and is observed right at the beginning of the spectral power-law
range. The overall behaviour is different for the three proxies, with the
total energy reaching a value of the exponent corresponding to uncorrelated,
random fluctuations, while both components reach slightly smaller $\kappa$.
Some degree of correlation is thus present in these two proxies, indicating
the presence of extremely fragmented current, vorticity, and Alfvénic
structures, whose superposition results in uncorrelated energy flux. The
magnetosheath sample is thus probably characterized by less developed
turbulence, corresponding to the steeper spectrum, and by the presence of
small-scale structures, in agreement with the large kurtosis.

Finally, in the WIND $1/f$ range, there is a similar trend as in the MMS-MS
data, with smooth fluctuations at large scale, but the increase to
uncorrelated, random values occurs sharply at the top of the inertial range,
so that the energy flow associated with large-scale fluctuations clearly does
not contribute to the energy cascade, as expected.

## Discussion

The nature of the turbulent energy cascade has been analyzed in three samples
of space plasmas by means of cancellation analysis applied to heuristic
proxies of the local energy transfer. The analysis provided information on the
sign alternation of the local mixed third-order fluctuations, which may be
related to the fractal properties of the associated energy transfer and thus
to small-scale dissipative processes. In two samples, namely in the solar wind
and in the KH instability at the magnetospheric boundary layer, the turbulent
cascade is well described by the proxies, and cancellation analysis captures
the increasing complexity of the alternating positive and negative
fluctuations. In these two samples, the energy is transferred to small scales
eventually generating disrupted current and vorticity structures, as well as
Alfvénic structures. The fractal dimension of these structures, obtained from
the cancellation exponents, is indicative of a strong concentration of energy
within a small fraction of the volume, typical of intermittency. The
cancellation analysis of the magnetosheath sample studied in this work, on the
contrary, provides an overall lower complexity estimate, which suggests the
presence of less evolved turbulence, and lack of formation of well-structured
energy channels. This is in agreement with the steeper spectrum and the more
irregular Politano-Pouquet scaling law.

These results help to characterize the fluctuations that carry energy to
smaller scales and provide the input or trigger for the activation of kinetic,
dissipative processes in the small-scale range Chen et al. (2019); Klein et
al. (2016); Sorriso-Valvo et al. (2019). Moreover, the estimated one-
dimensional projected fractal dimension provides information on the topology
of the different types of fluctuations, namely of the current sheets,
vorticity structures and Alfvénic fluctuations, that play an important role in
the dissipation of the turbulence.

###### Acknowledgements.

We acknowledge useful discussions with Denis Kuzzay, Lorenzo Matteini and Olga
Alexandrova. FF acknowledges HPC@POLITO for computational resources. LS and CV
were supported by EPN Internal Project PII-DFIS-2019-01. FF was supported by
grant FOIFLUT 37/17/F/AR-B. EY was supported by the Swedish Contingency
Agency, grant 2016-2102. CC was supported by STFC Ernest Rutherford Fellowship
ST/N003748/2. AG, KC, OK, DK and LS have received support from the Shota
Rustaveli National Science Foundation Project No FR17$\\_$279\. SP has
received funding from the European Unions Horizon 2020 Research and Innovation
programme under grant agreement No 776262 (AIDA, www.aida-space.eu). The
French LPP involvement for the SCM instrument on MMS was supported by CNRS and
CNES. MMS data are available at the MMS Science Data Center
(https://lasp.colorado.edu/mms/sdc). Wind data are available at CDAWeb
(https://cdaweb.gsfc.nasa.gov).

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