11institutetext: Swedish Institute of Space Physics (IRF), Ångström
Laboratory, Lägerhyddsvägen 1, 75121 Uppsala, Sweden 22institutetext: CNR,
Istituto per la Scienza e la Tecnologia dei Plasmi, via Amendola 122/D, 70126,
Bari, Italy 33institutetext: Space and Plasma Physics, School of Electrical
Engineering and Computer Science, KTH Royal Institute of Technology,
Teknikringen 31, SE-11428 Stockholm, Sweden 44institutetext: Université de
Lyon, CNRS, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon
1, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR5509, F-69134
Écully, France 55institutetext: Dipartimento di Scienze Fisiche e Chimiche,
Università degli Studi dell’Aquila, Via Vetoio 42, I-67100 Coppito AQ, Italy
66institutetext: National Institute for Astrophysics (INAF) -– Institute for
Space Astrophysics and Planetology (IAPS), Via Fosso del Cavaliere, 100, 00133
Rome, Italy 77institutetext: National Institute for Astrophysics (INAF) –
Astrophysical Observatory of Torino, Via Osservatorio 20, I-10025 Pino
Torinese, Italy

# Helios 2 observations of solar wind turbulence decay in the inner
heliosphere

L. Sorriso-Valvo, Corresponding author<EMAIL_ADDRESS>R. Marino
44 R. Foldes 4455 E. Lévêque 44 R. D’Amicis 66 R. Bruno 66 D. Telloni 77 E.
Yordanova 11

(Received September 15, 1996; accepted March 16, XXX)

###### Abstract

Aims. The linear scaling of the mixed third-order moment of the
magnetohydrodynamic fluctuations is used to estimate the energy transfer rate
of the turbulent cascade in the expanding solar wind.

Methods. In 1976 the Helios 2 spacecraft measured three samples of fast solar
wind originating from the same coronal hole, at different distance from the
sun. Along with the adjacent slow solar wind streams, these represent a unique
database for studying the radial evolution of turbulence in samples of
undisturbed solar wind. A set of direct numerical simulations of the MHD
equations performed with the Lattice-Boltzmann code FLAME is also used for
interpretation.

Results. We show that the turbulence energy transfer rate decays approximately
as a power law of the distance, and that both the amplitude and decay law
correspond to the observed radial temperature profile in the fast wind case.
Results from magnetohydrodynamic numerical simulations of decaying
magnetohydrodynamic turbulence show a similar trend for the total dissipation,
suggesting an interpretation of the observed dynamics in terms of decaying
turbulence, and that multi-spacecraft studies of the solar wind radial
evolution may help clarifying the nature of the evolution of the turbulent
fluctuations in the ecliptic solar wind.

###### Key Words.:

(Sun:) solar wind – Turbulence – magnetohydrodynamics MHD)

##  1 Introduction

Spacecraft observations of interplanetary fields and plasma show that the
solar wind is highly turbulent (Bruno & Carbone, 2013). After the onset of the
turbulent cascade at coronal level (Kasper et al., 2021; Bandyopadhyay et al.,
2022; Zhao et al., 2022), several processes may energize the fluctuations
during the solar-wind expansion (Verscharen et al., 2019): nonlinear decay of
large-scale Alfvén waves of solar or coronal origin (Malara et al., 2000;
Chandran, 2018), expansion-related and coronal-driven shears (Velli et al.,
1990; Tenerani & Velli, 2017), pick-up ions interaction, magnetic switchbacks
(Bale et al., 2021; Hernández et al., 2021; Telloni et al., 2022), large-scale
structures and instabilities (Roberts et al., 1992; Kieokaew et al., 2021).
The properties of turbulence are strongly variable (Bruno & Carbone, 2013),
reflecting the diversity of solar coronal sources, which modulate density,
velocity, temperature and ion composition of the plasma (von Steiger et al.,
2000). Solar wind intervals are often classified according to their bulk
speed, $V_{sw}$, as fast ($V_{sw}\gtrsim$ 600 km s-1) or slow
($V_{sw}\lesssim$ 500 km s-1). However, turbulence properties more clearly
depend on the Alfvénic nature of the fluctuations, for example measured using
the normalized cross-helicity,
$\sigma_{c}=\langle\delta\mathbf{v}\cdot\delta\mathbf{b}\rangle/\langle|\delta\mathbf{v}|^{2}+|\delta\mathbf{b}|^{2}\rangle$,
where the magnetic field $\mathbf{B}$ is transformed in velocity units through
the mass density $\rho$, $\mathbf{b}=\mathbf{B}/(4\pi\rho)^{1/2}$ (Matthaeus &
Goldstein, 1982), $\delta$ indicates fluctuations with respect to the mean,
and brackets indicate sample average. For example, large-scale Alfvénic
fluctuations may reduce the nonlinear energy transfer by sweeping apart the
interacting structures (Kraichnan, 1965; Dobrowolny et al., 1980).

In non-Alfvénic solar wind (often observed in slow intervals), the turbulence
generates broadband power-law magnetic spectra, $E(f)\sim f^{-\alpha}$.
Scaling exponents $\alpha$, close to Kolmogorov’s 5/3 (Kolmogorov, 1941), are
observed from the injection scales of solar wind structure ($\sim$hours), to
the characteristic ion scales ($\sim$10 s at 1 au) (Bruno & Trenchi, 2014),
where field-particle effects become relevant and the spectral exponents
increase (Leamon et al., 1998). Within such broad inertial range, strong
intermittency is observed (Sorriso-Valvo et al., 1999), revealing
inhomogeneously distributed small-scale structures, such as vortices and
current sheets, generated by the nonlinear interactions (Salem et al., 2009;
Greco et al., 2016). These characteristics are robustly observed at any
distance from the Sun beyond 0.3 au, with no or limited radial evolution of
spectral range extension or intermittency (Bruno et al., 2003).

On the other hand, in typically fast Alfvénic wind strongly aligned low-
frequency velocity and magnetic fluctuations produce a $1/f$ magnetic spectral
range (Matthaeus & Goldstein, 1986; Verdini et al., 2012; Chandran, 2018;
Matteini et al., 2018). The classical turbulence inertial range is narrower,
with spectral exponents between 5/3 and the Kraichnan’s 3/2 (Kraichnan, 1965).
Note that Alfvénic slow solar wind was recently abundantly observed close to
the Sun (Chen et al., 2020), and less frequently near 1 au (D’Amicis et al.,
2021). In the Alfvénic wind, turbulence shows a clearer evolution as the wind
radially expands, with the $1/f$ break drifting towards lower frequencies
(Bavassano et al., 1982b). The inertial range broadening is interpreted as the
growing Reynolds’ number (Frisch, 1995; Parashar et al., 2019), which together
with the observed increasing intermittency (Bruno et al., 2003, 2014), would
suggest an evolution of the solar wind dynamics towards more developed
turbulence states. (Tu & Marsch, 1995; Bavassano et al., 2002; Burlaga, 2004;
Macek et al., 2012; Bruno & Carbone, 2013; Fraternale et al., 2016; Chen et
al., 2020; Bandyopadhyay et al., 2020). Such evolution corresponds to the
observed gradual decrease of the Alfvénic alignment between velocity and
magnetic fluctuations (Bavassano et al., 1998, 2002). Alternatively,
observations could result from the competing action between a coherent
component (the intermittent inertial range structures generated by turbulence)
with a stochastic component ($1/f$-range propagating Alfvénic fluctuations,
Bruno et al., 2001, 2003; Borovsky, 2008). In both interpretations, the slow
solar wind milder evolution is therefore associated with the reduced presence
of Alfvénic fluctuations (Tu & Marsch, 1995; Bruno & Carbone, 2013). Recent
observations of Alfvénic solar wind closer to the Sun confirmed the radial
evolution properties briefly described above (Chen et al., 2020; Bourouaine et
al., 2020; Hernández et al., 2021). The above observations of spectra and
intermittency are generally used to support the evolving nature of the solar
wind turbulence in the inner heliosphere, and to constrain global solar wind
models and their energy budget.

As an alternative to spectra, the turbulent cascade can be examined using the
scaling properties of the third-order moments of the fluctuations (Marino &
Sorriso-Valvo, 2023). Based on robust theoretical predictions (Politano &
Pouquet, 1998), third-order laws allow to estimate the energy transfer rate of
turbulence (Sorriso-Valvo et al., 2007). Under specific assumptions, this
quantity represents a more fundamental measure of the state of turbulence, in
comparison to power spectra and intermittency. Furthermore, studies of radial
evolution of turbulence mostly rely on a statistical approach, so that
different samples may include plasma from different originating coronal
regions or solar activity, resulting in different initial energy injection,
nonlinear coupling efficiency, or in-situ energy injection from instabilities
or large-scale structures. These can all diversely contribute determining the
properties and energetic content of the turbulent cascade. One possible way to
mitigate such inhomogeneity is to measure the same plasma with two spacecraft
that are occasionally radially aligned at different distance from the sun
(D’Amicis et al., 2010; Telloni et al., 2021a). Alternatively, under optimal
orbital configurations, plasma from a steady solar source can be measured by
the same spacecraft at different times (see, e.g., Bruno et al., 2003).

In this article, we study the status of the turbulence at different radial
distances from the Sun using the third-order moment law and intermittency, as
measured in a set of recurrent streams of solar wind measured by the Helios 2
spacecraft. Based on a qualitative comparison with the plasma generated with a
magnetohydrodynamic simulation in the spin-down phase, we interpret the
observed trend of the energy transfer rate in the solar wind as to be
determined, among other process, by a decay of turbulence occurring with the
heliocentric distance. In Sections 2 and 3 we describe the dataset and the
methodology used to investigate the status of turbulence. Section 4 provides
the results of the analysis of Helios 2 data, while in Section 5 results of a
lattice-Boltzmann numerical simulations are shown. Finally, conclusions are
given in Section 6.

##  2 Helios 2 Data

One popular case of measurements of solar wind from the same source occurred
in 1976, when Helios 2 measured plasma and fields of three fast solar wind
streams at different distances (0.9, 0.7 and 0.3 au) originated from a
persistent polar coronal hole, which was reasonably stable during nearly two
solar rotations (Bavassano et al., 1982a; Bruno et al., 2003). Similarly,
three preceding slow, non Alfvénic wind streams were used as samples of
evolving slow solar wind. These streams have provided outstanding information
about the radial evolution of turbulence, since the initial conditions were
statistically steady and no stream interactions were included, Note, however,
that the more dynamical solar source of the slow streams might not be as
steady as in the case of the coronal hole generating the fast wind. Figure 1
shows an overview of solar wind bulk speed $V_{sw}$, spacecraft radial
distance from the sun $R$, proton density $n_{p}$, and magnetic field
magnitude $B$ during the days 45 to 110 of 1976. Six color-shaded areas
identify the selected streams, at 0.3 au (red), 0.7 au (green) and 0.9 au
(blue). For each distance, lighter colors and dashed lines indicate slow
streams, darker colors and full lines identify fast streams.

Figure 1: Helios 2 measurements during days 45—110 of 1976. From top to
bottom: solar wind bulk speed $V_{sw}$, distance from the sun $R$, proton
density $n_{p}$, and magnetic field magnitude $B$. Colored shaded areas
identify the selected streams at 0.3 au (red), 0.7 au (green) and 0.9 au
(blue). Lighter (darker) colors indicate slow (fast) streams.

##  3 Methodology: the Politano-Pouquet law

Past studies of turbulence mainly relied on spectral and structure functions
analysis (Bavassano et al., 1982a, b; Bruno et al., 1985; Bavassano et al.,
2002; Bruno et al., 2003, 2004, 2004, 2014; Perrone et al., 2018). However,
the nature of the turbulent cascade is more thoroughly captured by the scaling
of the third-order moments of the fluctuations, an exact relation obtained
from the incompressible MHD equations under the hypothesis of stationarity,
isotropy, and large Reynolds’ number (Politano & Pouquet, 1998). Called
Politano-Pouquet (PP) law, it prescribes that the mixed third-order moment of
the MHD fields fluctuations is a linear function of the scale:

$\displaystyle Y(\Delta t)$ $\displaystyle\equiv$ $\displaystyle\langle\Delta
v_{L}(|\Delta\mathbf{v}|^{2}+|\Delta\mathbf{b}|^{2})-2\Delta
b_{L}(\Delta\mathbf{v}\cdot\Delta\mathbf{b})\rangle$ (1) $\displaystyle=$
$\displaystyle\frac{4}{3}\varepsilon V_{sw}\Delta t\,.$

$\Delta\phi_{L}$ are longitudinal increments of the component of a generic
scalar or vector component $\phi$ in the sampling direction, with the magnetic
field $\mathbf{B}$ in velocity units through the mass density $\rho$,
$\mathbf{b}=\mathbf{B}/(4\pi\rho)^{1/2}$. The mean bulk speed $V_{sw}$ allows
transforming spatial lags $\Delta l$ to temporal lags $\Delta t$ via the
Taylor hypothesis, $\Delta l=-V_{sw}\Delta t$ (Taylor, 1938). Observing linear
scaling ensures that a turbulent cascade is active and fully developed, and
that the complex hierarchy of structures on all scales is well formed and
sustains the cross-scale energy transfer leading to small-scale dissipation.
Measuring the energy transfer rate $\varepsilon$ provides a quantitative
estimate of the turbulent energy flux. This is an invaluable information in
the collisionless solar wind, where energy dissipation cannot be measured
using viscous-resistive modeling. The same quantity could in principle be
obtained from the (Kolmogorov) spectrum, but its evaluation includes a
constant factor that can hardly be obtained in the highly variable solar wind.
Third-order scaling laws (Marino & Sorriso-Valvo, 2023) have been used to
determine the properties of turbulence in the solar wind at 1 au (MacBride et
al., 2005; Smith et al., 2009; Stawarz et al., 2010; Coburn et al., 2012), in
the outer (Sorriso-Valvo et al., 2007; Marino et al., 2008; Carbone et al.,
2009; Marino et al., 2012) and inner (Gogoberidze et al., 2013; Bandyopadhyay
et al., 2020; Hernández et al., 2021; Wu et al., 2022) heliosphere, and in
near-Earth (Hadid et al., 2017; Quijia et al., 2021) and near-Mars (Andrés et
al., 2020) environments. Data from Ulysses in the polar outer heliosphere
(Marino et al., 2008, 2012; Watson et al., 2022) and from Parker Solar Probe
in the ecliptic inner heliosphere (Bandyopadhyay et al., 2020) suggest that
the energy transfer rate statistically decreases radially. At the same time,
the fraction of solar wind intervals where the linear scaling is observed
increases radially (Marino et al., 2012), in agreement with the observed
decrease of the cross-helicity and generally supporting the evolving nature of
the turbulence in the expanding heliosphere.

##  4 Results: mixed third-order moment scaling laws, turbulent energy
transfer rate and intermittency

Here we present the analysis of the PP scaling law in the three fast and three
slow streams described above, using the 81 s resolution plasma and magnetic
field measured by Helios 2. The mixed third-order moments, Equation (1), are
displayed in Figure 2 for each of the six intervals. Statistical convergence
of the samples was tested using standard techniques (Dudok De Wit, 2004;
Kiyani et al., 2006). For both fast and slow streams, the different colors
indicate different distances, as stated in the legend. The mixed third-order
moments $Y$ are mostly positive (full symbols), indicating direct energy
transfer form large to small scales. Negative values, most probably due to
lack of convergence or presence of large-scale velocity shears (Stawarz et
al., 2011), are indicated by open symbols, and are not considered in this
study.

Figure 2: PP scaling for fast (top) and slow (bottom) streams, at three
different distances from the sun. Linear fits are indicated (grey lines),
along with the mean energy transfer rate, $\varepsilon$ (kJ kg-1 s-1) (color
coded). Full and empty symbols refer to positive and negative values of $Y$,
respectively.

An inertial range was identified for each case, at timescales between 81 s and
$\sim$20 minutes, although the linear scaling is better defined and more
extended in the samples at 0.3 au. The upper scales observed here are larger
than, but roughly consistent with, the outer scale of the turbulence,
estimated as the correlation timescale, $\tau_{c}$ (Greco et al., 2012), and
partially include the $1/f$ range, as will be discussed below. This shows that
the six intervals can be considered as samples of fully developed turbulence.

Fitting the PP law provides the mean energy transfer rate, given in colors
next to each fitted line in Figure 2. The fit is performed on a range
including more than one decade of scales in all cases except for the slow
solar wind at 0.9 au, where a slightly shorter range is covered. It should be
observed that in the two examples studied here the linear scaling range is
broader closer to the sun, while the scaling becomes less clear near 1 au.

However, for the purpose of this study, the relevant information is the energy
transfer rate, which is sufficiently well represented by the power-law fits
shown in Figure 2. The first notable characteristic is that slow streams have
smaller energy transfer rate than fast streams at all distances, suggesting
that the initial energization of the turbulence is stronger in fast wind,
perhaps not systematically but in the cases under study, or that its decay is
faster in slow wind. This is consistent with the known correlation between
turbulence amplitude and both proton temperature and wind speed (see, e.g.,
Grappin et al., 1991). Furthermore, in both fast and slow wind the energy
transfer rate consistently decreases with the distance, as expected from the
observed decay of the turbulent fluctuations (Bruno & Carbone, 2013). Such
observations seem to indicate that in the fast streams, while the spectrum
broadens and the small-scale intermittent structures emerge (Bavassano et al.,
1982a; Bruno et al., 2003), the cascade transports less energy across the
scales. The same decreasing energy transfer is observed in the otherwise
steadier slow streams.

Figure 3: Top panel: energy transfer rate, $\varepsilon$, versus the distance
from the sun, $R$. Fast streams are indicated with dark purple squares, slow
streams with dark orange circles. Power-law fits and the relative scaling
exponents are shown. The heating rate obtained using equation (2),
$\varepsilon_{T}$, is also indicated with purple diamonds (fast streams) and
orange diamonds (slow wind). Bottom panel: the temperature decay for the fast
(cyan squares) and slow (green circles) streams, with power-law fits and
indicated exponents.

In the top panel of Figure 3, the energy transfer rate is plotted versus the
distance from the Sun. The measured values can be compared with standard
estimates of the proton heating rate (Marino et al., 2008, 2011). This can be
obtained through a simple model that, under the isotropic fluid approximation,
neglecting heat fluxes, and assuming a stationary flow, uses the solar wind
bulk speed and temperature radial profiles to determine the proton heating
rate accounting for deviation from adiabatic cooling (Verma, 2004; Vasquez et
al., 2007). In particular, if the proton temperature decays as a power law of
the distance, $T(R)\sim R^{-\xi}$, then the proton heating rate is given by

$\varepsilon_{T}=\frac{3}{2}\left(\frac{4}{3}-\xi\right)\frac{k_{B}V_{SW}(R)T(R)}{Rm_{p}}\,,$
(2)

where $k_{B}$ is the Boltzmann constant and $m_{p}$ is the proton mass. The
decay exponent $\xi$ of the temperature can be estimated using Helios 2 data.
For the two sets of fast and slow streams, the mean temperature for each
stream is shown in the bottom panel of Figure 3. Assuming power-law decay, a
fitting procedure provides $\xi=0.87\pm 0.01$ for the fast streams, and
$\xi=1.29\pm 0.03$ for the slow streams. These considerably deviate from the
expected adiabatic value ($\xi=4/3$) for the fast wind, while for the slow
streams the cooling is closer to adiabatic. Totten et al. (1995) used Helios 1
data to obtain radial profiles of the proton temperature, which was observed
to consistently decay with exponent $\xi\simeq 0.9$, independent of the wind
speed. Note that the results shown here might be specific to the case under
study, while in Totten et al. (1995) a large statistical sample was used.
Similar results were found using MHD numerical simulations (Montagud-Camps et
al., 2018, 2020). The Helios 2 results shown here are in good agreement with
the above observations and models for the fast wind streams, but not for the
slow ones.

Using the obtained temperature radial decay exponent and the mean speed and
temperature of the three streams, the radial profile of the approximate
heating rate can be estimated. The values obtained are shown in the top panel
of Figure 3. For the fast streams, the agreement with the estimated energy
transfer rate is excellent. This demonstrates that the energy transfer rate
estimated using the incompressible, isotropic version of the PP law are
sufficiently accurate. On the other hand, for the slow wind the required
heating is nearly one order of magnitude smaller than the observed turbulent
heating rate. This is a consequence of the very weak deviation from adiabatic
of the power-law temperature decay. Possible reasons for this discrepancy
include: (i) the relatively poor power-law profile of $T(R)$ and subsequent
underestimation of the required heating rate; (ii) the possible variability in
the slow wind source region, which would affect the stationarity assumption
for the model and result in unaccounted for temperature variability; (iii)
energy lost to heat flux and electron heating, not included in equation (2),
and which might be more relevant than for the Alfvénic, fast wind.
Nevertheless, even for the slow streams the decay of the predicted heating
rate is close to the observed decay of the turbulent energy transfer rate.

In fact, for plasma proceeding from an approximately stationary coronal
structure and far from stream interaction regions, the turbulent energy
transfer can be expected to decrease as a power law of the distance. If we
assume for the energy transfer rate a power-law radial decay, $\varepsilon\sim
R^{-\alpha}$, the measured values can be fitted to power-laws, providing the
decay exponents $\alpha_{F}\simeq 1.8\pm 0.2$ and $\alpha_{S}\simeq 2.3\pm
0.2$ for fast and slow streams, respectively. The slower decay observed in the
fast streams could be the result of the local energy injection from the $1/f$
reservoir, which may partially compensate the dissipation losses and thus
slightly slow down the decay in comparison with the slow wind, where no
supplementary injection is provided.

The above scenario provides a quantitative estimate of the turbulence decay
observed in undisturbed expanding samples, such as the Helios 2 recurrent
streams. Such observation could be useful to constrain models of the radial
evolution of turbulence. These may include, for example, the simple damping of
both velocity and magnetic fluctuations due to the conservation of angular
momentum and of magnetic flux in an expanding plasma volume advected by the
radial wind (Parker, 1965; Heinemann & Olbert, 1980), as well as local energy
injection as it results from expansion and large-scale shears (Velli et al.,
1990; Grappin et al., 1993; Tenerani & Velli, 2017), or modification of the
timescale of nonlinear interactions associated with the radial decrease of
Alfvénicity (Smith et al., 2009; Stawarz et al., 2010). Since the results
presented here are based on only two case-study events, they lack generality.
On the other hand, they do provide more rigorous parameters for the specific
intervals. A larger study of similar events, studied individually, is however
necessary to cover a broader range of parameters. In order to complete the
turbulence analysis of the streams under study, we quantify the degree of
intermittency of the turbulent fluctuations. Intermittency is typically
described by the scaling of the statistics of the fields fluctuations,
measured through probability distribution functions (Sorriso-Valvo et al.,
1999; Pagel & Balogh, 2003), structure functions (Carbone, 1994; Kiyani et
al., 2009), or multifractal analysis (Macek et al., 2012; Alberti et al.,
2019). A standard estimator of intermittency is provided by the power-law
scaling of the kurtosis of a magnetic field component $B_{i}$, $K(\Delta
t)=\langle\Delta B_{i}^{4}\rangle/\langle\Delta B_{i}^{2}\rangle^{2}\sim\Delta
t^{-\kappa}$. The scaling exponent, $\kappa$, provides a quantitative measure
of intermittency, being related to the fractal co-dimension of the
intermittent structures (Castaing et al., 1990; Carbone & Sorriso-Valvo, 2014;
Sorriso-Valvo et al., 2015). For these intervals, the kurtosis was already
presented by Bruno et al. (2003). Nevertheless, here we perform a more
detailed, quantitative study that will provide additional description of the
intermittency. The two panels of Figure 4 show the magnetic field kurtosis for
the three fast (top panel) and slow (bottom panel) solar wind streams,
computed using 6-second cadence magnetic vectors and averaging over the three
field components (after verifying that all individual components displayed
similar behaviour). In the fast streams, two power-law scaling ranges are
identified in the inertial range, $\sim$6—200 s, and at lower frequency,
$\sim$200—8000 s. In agreement with spectral observations, the break between
the two ranges migrates towards larger scales with increasing distance from
the Sun (see, e.g., Bruno & Carbone, 2013, and references therein). Note that,
for the streams at 0.3 and 0.7 au, the low-frequency range mostly includes the
$1/f$ spectral range. Power-law fits provide the scaling exponents $\kappa$,
which are indicated in the figure. The inertial-range exponents agree with
previous observations (Di Mare et al., 2019; Sorriso-Valvo et al., 2021),
while the smaller $1/f$-range values are closer to fluid turbulence’s (where
exponents are typically around 0.1; see, e.g., Anselmet et al., 1984). In both
ranges, the values of $K$ and the scaling exponents quantitatively confirm the
radial increase of intermittency (Bruno et al., 2003). To the best of our
knowledge, the kurtosis’ power-law scaling in the $1/f$ range was not observed
before. Jointly with the observation of the PP law discussed above, it
suggests that, at least in the streams closer to the Sun, nonlinear
interactions are effectively transferring energy across scales, even in the
$1/f$ range. This observation opens interesting questions about the nature of
the fluctuations in the low-frequency range, which calls for more detailed
studies. On the other hand, in the slow wind streams, where no $1/f$ spectral
range is observed, a single power-law covers the whole range, with no clear
radial trend. The scaling exponents reveal strong intermittency at all
distances.

Figure 4: Magnetic field kurtosis, $K$, for fast streams (top) and slow
streams (bottom), averaged over the field components. Power-law fits in the
5/3 and $1/f$ spectral ranges are indicated with the fitted scaling exponents
(fitting errors are $\leq 0.01$).

##  5 Comparison with numerical simulations of unforced MHD turbulence

The power-law radial decay of the turbulent energy transfer rate and the
associated increase of intermittency, highlighted in the previous section,
represent a solid constraint for models of solar wind turbulence. It is
interesting to notice that similar properties are also typical features of
unforced fluid and magneto-fluid turbulence, as observed in numerical
simulations (Biskamp, 1993; Hossain et al., 1995; Miura, 2019; Bandyopadhyay
et al., 2019). Using a decaying direct numerical simulation integrating the
incompressible MHD equations, with no mean magnetic field (Foldes et al.,
2022), here we show how a simplified framework not incorporating signature
features of solar wind turbulence (for example anisotropy), not accounting for
effects of the solar wind expansion (Grappin et al., 1993; Verdini & Grappin,
2015; Tenerani & Velli, 2017), is able to qualitatively reproduce trends and
statistics observed by Helios, thus can be used to decipher the phenomenology
underlying the turbulent energy transfer in the solar wind. What emerged from
our analysis is that the phenomenology described in the previous sections is
compatible with the temporal decay of a magnetohydrodynamic unforced plasmas.
In other words, the basic three-dimensional MHD simulations used here provide
indications of qualitative similarity between the radial evolution of
turbulence in the expanding solar wind and the temporal decay of unforced MHD
turbulence via viscous-resistive dissipation.

run | $\mathrm{Re=Re_{m}}$ | $v_{rms}^{*}$ | ${v_{rms}^{0}}/{b_{rms}^{0}}$ | $\mathrm{Ma}$ | $t^{*}$ | $\xi$
---|---|---|---|---|---|---
I |  $\sim$ 1500 | $8.5\cdot 10^{-4}$ | $\sim 0.5$ | 0.005 | 79.5 | 2.4
II |  $\sim$ 900 | $2.7\cdot 10^{-3}$ | $\sim 1.0$ | 0.005 | 51.9 | 1.7
III |  $\sim$ 800 | $2.5\cdot 10^{-3}$ | $\sim 2.0$ | 0.001 | 45.3 | 1.8

Table 1: Adimensional parameters at the current peak $t^{*}$ (in turnover time
units). $\mathrm{Re}$ and $\mathrm{Re_{m}}$ are respectively the Reynolds and
magnetic Reynolds numbers; $\mathrm{Ma}$ is the Mach number;
$v_{rms}^{0}/b_{rms}^{0}$ is the ratio between the $rms$ velocity and magnetic
fluctuations at the initial time of the simulation ($t=0$); $\xi$ is the
fitted exponent (see Figure 6). Box size, $(2\pi)^{3}$, and magnetic Prandtl
number, $Pr_{m}=1$, are the same for all runs.

This is not in contrast with the expanding-box simulations previously
suggesting that the inclusion of expansion effects results in faster energy
decay and in inverted cross helicity radial profile, that switches from
increasing to decreasing with distance (see, e.g., Dong et al., 2014;
Montagud-Camps et al., 2020, 2022). In this work, we use actually a lattice
Boltzmann (LB) code, FLAME (Foldes et al., 2022), to integrate the quasi-
incompressible MHD equations in a three-dimensional periodic domain, not
expanding. In the LB approach, the volume discretization is operated on a gas
of particles distributed on a lattice, rather than on a grid. The dynamics of
the particles develops in the frame of the kinetic theory, the temporal
evolution of the plasma being achieved through the recursive application of
simple collision and streaming operations. The macroscopic MHD fields (e.g.
fluid velocity $\mathbf{v}$, density $\rho$, and magnetic field $\mathbf{B}$)
are then obtained through integration of the statistical moments of the
particle distribution functions. Please note that the particles here are not
plasma particles, they exist at the level of the numerical scheme and are
instrumental to the LB approach to obtain (in the case of FLAME) the fields of
the simulated magnetohydrodynamic plasma (see details in Foldes et al., 2022).
We examine three runs, whose parameters are listed in Table 1. The latter are
initialized with a standard Orszag-Tang (OT) vortex (Orszag & Tang, 1979;
Mininni et al., 2006), using different kinetic-to-magnetic energy ratio,
$v_{rms}^{0}/B_{rms}^{0}$, where $v_{rms}^{0}$ and $B_{rms}^{0}$ are the
initial rms values of the field fluctuations (for the Helios 2 solar wind
intervals studied here, $v_{rms}/B_{rms}\sim$ 0.5–1, Bruno et al., 1985). The
integration is performed over a $512^{3}$-point three-dimensional lattice
(Orszag & Tang, 1979; Mininni et al., 2006), the functional form of OT being:
$\mathbf{U}(\mathbf{x},0)=v_{rms}^{0}\left[-2sin(y),2sin(x),0\right]$ and
$\mathbf{B}(\mathbf{x},0)=B_{rms}^{0}\left[-2sin(2y)+sin(z),2sin(x)+sin(z),sin(x)+sin(y)\right]$.
Before computing any statistics, the simulation is let to evolve until the
plasma reaches a state of fully developed turbulence when the volume averaged
density current ,$\langle j\rangle$, attains its peak value, at the time
$t^{*}$. A snapshot of the vorticity field at $t^{*}$ is shown for run I in
Figure 5, along with the corresponding isotropic magnetic and kinetic spectral
trace.

Figure 5: Left panel: rendering of the vorticity field, at the time $t^{*}$
when the peak of the density current is reached. Right panel: isotropic
kinetic and magnetic energy spectra for run I in the table I of the main text
at $t^{*}$.

For all runs, spectra exhibit the typical extended power-law inertial ranges,
which enables qualitative comparison to the SW.

It must be pointed out that, since the solar wind speed is reasonably steady,
the radial distance of each of the six Helios 2 streams analyzed here can be
ideally converted to time of travel from an arbitrary initial position where
the turbulence peaks, $R_{0}$. We can expect this to be close to the Sun
(Bandyopadhyay et al., 2020), for example at the Alfvén point (Kasper et al.,
2021; Zhao et al., 2022), or it can be identified with the stream under study
closest to the Sun ($R_{0}=0.3$ au). Further expressing the solar wind time of
travel in units of the initial nonlinear time, $t_{NL}$ (here taken as the
average between characteristic kinetic and magnetic nonlinear times,
respectively $t_{v}=L_{int}^{v}/v_{rms}^{*}$ and
$t_{B}=L_{int}^{B}/B_{rms}^{*}$, where $L_{int}^{v}$ and $L_{int}^{B}$ are the
kinetic and magnetic integral scales, whereas $v_{rms}^{*}$ and $B_{rms}^{*}$
are the rms values of the field fluctuations, all estimated at $R_{0}$)
enables the comparison between observational and numerical estimates and
statistics. This suggests the possible use of the following simple expression
to determine the normalized “age” of the turbulence:
$t_{turb}=(R-R_{0})/(V_{sw}t_{NL})$. We will not make use of this parameter
since our comparison with the numerical simulation results is only
qualitative. Furthermore, determining $R_{0}$ and the integral scales is not
trivial in solar wind data. However, since the parameters in the above
transformation (the solar wind speed, the initial distance, and the nonlinear
time at the initial position) can all be considered as constant, the power-law
scaling presented will not be affected by this transformation. Computing
spectral energy fluxes from numerical simulation requires the integration of
quantities over extended intervals, during which the system is assumed to be
stationary. This condition can hardly be attained when dealing with spin-down
runs. Moreover we would like to monitor the evolution of the turbulent energy
transfer rate, in all the simulations in Table 1, at different times after the
plasma has reached the density current peak. For this reason, here we make the
assumption (reasonable in case of fully developed turbulence) that the rate at
which kinetic and magnetic energies are transferred throughout the inertial
range, in subsequent time intervals, as the system relaxes due to viscous
effects, does follow the same trend of the small-scale dissipation with the
evolutionary time of the simulations. We have thus computed systematically the
volume-averaged total dissipation rate,
$\varepsilon_{tot}=\varepsilon_{V}+\varepsilon_{B}$, where
$\varepsilon_{V}=\nu\langle({\nabla v})^{2}\rangle$ and
$\varepsilon_{B}=\eta\langle j^{2}\rangle$ are the kinetic and magnetic
dissipation, respectively, and $\nu=\eta$ are the kinematic viscosity and the
resistivity. Figure 6 (top panel) displays the temporal evolution of the
dissipation rate $\varepsilon_{tot}$ in runs I-III, starting from the
turbulence peak $t^{*}$. All times are expressed in units of the nonlinear
time, $t_{NL}$, estimated as described above using the simulation parameter
computed at the time of the peak of the density current, $t^{*}$. For all
runs, a power-law time evolution of the energy transfer rate can be clearly
identified (Batchelor et al., 1948; Hossain et al., 1995), with fitted scaling
exponents compatible with those observed in the solar wind.

Figure 6: Top panel: energy dissipation rate, $\varepsilon_{tot}$ (arbitrary
units), versus the rescaled simulation time, $(t-t^{*})/t_{NL}$, for the three
simulation runs. Bottom panel: magnetic field kurtosis, $K$, versus the
spatial increment $\Delta l$, averaged over the field components, shown here
for run II at three different times (see legend). For all plots, power-law
fits and the relative fitted exponents are indicated (fitting errors $\leq
0.01$).

Finally, the bottom panel of Figure 6 shows the magnetic field kurtosis versus
the scale $\Delta l$ for Run II, at three different times $t>t^{*}$ in the
simulation. The power-law scaling exponents indicate increasing intermittency
with time (Biferale et al., 2003), similarly to observations in the fast
recurrent stream studied here. The same trend is observed for all runs.

By qualitatively comparing both the energy transfer rate and intermittency,
evident similarities arise between the time evolution in our simulations of
decaying MHD turbulence and the radial profile in the solar wind streams.
Although the numerical model used here is not intended to fully reproduce the
solar wind features, such similarities suggest that the ongoing dissipation of
turbulent fluctuations could concur to determine the observed radial evolution
of solar wind turbulence.

##  6 Conclusions

We used Helios 2 measurements of the solar wind emitted from a steady coronal
source and without interactions with coronal or heliospheric structures,
collected at different distances from the Sun. We have shown that the
turbulence energy transfer rate decays approximately as a power law of the
distance, and we provided measured decay exponents that may be used to
constrain models of solar wind expansion. It should be pointed out that the
linear scaling range becomes narrow at larger distance from the Sun. In the
slow wind, for example, such range covers slightly less than a decade, which
is in contrast with the observed broad spectral inertial range. This is likely
due to the limited statistics provided by the relatively low resolution Helios
data, and more in general to the difficult observation of signed third-order
scaling laws. Possible other reasons include the violation of the isotropy
assumption and the presence of large-scale inhomogeneities (Stawarz et al.,
2011; Verdini et al., 2015). In the Alfvénic fast wind, the turbulence decay
is also associated with increasing intermittency. The observations presented
here are qualitatively compared with three-dimensional direct numerical
simulations of decaying MHD turbulence. Despite the simulations used here do
not include important elements such as the radial expansion of the solar wind,
the observed similarity between trends of energy transfer and dissipation
rates (estimated from Helios 2 observations and DNS, respectively) supports
the possible relevance of dissipation in the radial evolution of solar wind
turbulence. Furthermore, the observation in the $1/f$ range of both the PP law
and power-law scaling of the kurtosis suggests that, in fast solar wind, a
turbulent cascade is active also at large scales, even in the presence of
strongly Alfvénic large-scale fluctuations (Verdini et al., 2012). The
behavior highlighted by our analysis, together with the observed parameters,
can be relevant to constrain models of turbulence in the expanding solar wind
and of the plasma heating observed in both fast and slow streams (Marino et
al., 2008, 2011). Coordinated studies of PSP and Solar Orbiter measurements
will add statistical significance to our observations (Velli et al., 2020;
Telloni et al., 2021b).

###### Acknowledgements.

L.S.-V. and E.Y. were supported by SNSA grants 86/20 and 145/18. L.S.-V. is
supported by the Swedish Research Council (VR) Research Grant N. 2022-03352.
R.M. and R.F. acknowledge support from the project “EVENTFUL”
(ANR-20-CE30-0011), funded by the French “Agence Nationale de la Recherche” -
ANR through the program AAPG-2020. This research was supported by the
International Space Science Institute (ISSI) in Bern, through ISSI
International Team project #556 (Cross-scale energy transfer in space
plasmas).

## References

  * Alberti et al. (2019) Alberti, T., Consolini, G., Carbone, V., et al. 2019, Entropy, 21
  * Andrés et al. (2020) Andrés, N., Romanelli, N., Hadid, L. Z., et al. 2020, ApJ, 902, 134
  * Anselmet et al. (1984) Anselmet, F., Gagne, Y., Hopfinger, E. J., & Antonia, R. A. 1984, J. Fluid Mech., 140, 63
  * Bale et al. (2021) Bale, S. D., Horbury, T. S., Velli, M., et al. 2021, ApJ, 923, 174
  * Bandyopadhyay et al. (2020) Bandyopadhyay, R., Goldstein, M. L., Maruca, B. A., et al. 2020, ApJS, 246, 48
  * Bandyopadhyay et al. (2022) Bandyopadhyay, R., Matthaeus, W. H., McComas, D. J., et al. 2022, ApJ, 926, L1
  * Bandyopadhyay et al. (2019) Bandyopadhyay, R., Matthaeus, W. H., Oughton, S., & Wan, M. 2019, Journal of Fluid Mechanics, 876, 5–18
  * Batchelor et al. (1948) Batchelor, G. K., Townsend, A. A., & Taylor, G. I. 1948, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 193, 539
  * Bavassano et al. (1982a) Bavassano, B., Dobrowolny, M., Fanfoni, G., Mariani, F., & Ness, N. 1982a, Solar Phys., 78, 373
  * Bavassano et al. (1982b) Bavassano, B., Dobrowolny, M., Mariani, F., & Ness, N. 1982b, J. Geophys. Res., 87, 3617
  * Bavassano et al. (1998) Bavassano, B., Pietropaolo, E., & Bruno, R. 1998, J. Geophys. Res., 103, 6521
  * Bavassano et al. (2002) Bavassano, B., Pietropaolo, E., & Bruno, R. 2002, Journal of Geophysical Research: Space Physics, 107, 7
  * Biferale et al. (2003) Biferale, L., Boffetta, G., Celani, A., et al. 2003, Physics of Fluids, 15, 2105
  * Biskamp (1993) Biskamp, D. 1993, Magnetohydrodynamic Turbulence (Cambridge University Press)
  * Borovsky (2008) Borovsky, J. E. 2008, Journal of Geophysical Research: Space Physics, 113
  * Bourouaine et al. (2020) Bourouaine, S., Perez, J. C., Klein, K. G., et al. 2020, ApJ, 904, L30
  * Bruno et al. (1985) Bruno, R., Bavassano, B., & Villante, U. 1985, J. Geophys. Res., 90, 4373
  * Bruno & Carbone (2013) Bruno, R. & Carbone, V. 2013, Liv. Rev. in Solar Phys., 10, 2
  * Bruno et al. (2004) Bruno, R., Carbone, V., Primavera, L., et al. 2004, Annales Geophysicae, 22, 3751
  * Bruno et al. (2003) Bruno, R., Carbone, V., Sorriso-Valvo, L., & Bavassano, B. 2003, J. of Geophys. Res.: Space Phys., 108, 1130
  * Bruno et al. (2001) Bruno, R., Carbone, V., Veltri, P., Pietropaolo, E., & Bavassano, B. 2001, Planet. Space Sci., 49, 1201
  * Bruno et al. (2004) Bruno, R., Sorriso-Valvo, L., Carbone, V., & Bavassano, B. 2004, Europhysics Letters (EPL), 66, 146
  * Bruno et al. (2014) Bruno, R., Telloni, D., Primavera, L., et al. 2014, ApJ, 786, 53
  * Bruno & Trenchi (2014) Bruno, R. & Trenchi, L. 2014, ApJ, 787, L24
  * Burlaga (2004) Burlaga, L. F. 2004, Nonlinear Processes in Geophysics, 11, 441
  * Carbone & Sorriso-Valvo (2014) Carbone, F. & Sorriso-Valvo, L. 2014, Eur. Phys. J. E, 37, 61
  * Carbone (1994) Carbone, V. 1994, Annales Geophysicae, 12, 585
  * Carbone et al. (2009) Carbone, V., Marino, R., Sorriso-Valvo, L., Noullez, A., & Bruno, R. 2009, Phys. Rev. Lett., 103, 061102
  * Castaing et al. (1990) Castaing, B., Gagne, Y., & Hopfinger, E. J. 1990, Physica D Nonlinear Phenomena, 46, 177
  * Chandran (2018) Chandran, B. D. G. 2018, Journal of Plasma Physics, 84, 905840106
  * Chen et al. (2020) Chen, C. H. K., Bale, S. D., Bonnell, J. W., et al. 2020, ApJS, 246, 53
  * Coburn et al. (2012) Coburn, J. T., Smith, C. W., Vasquez, B. J., Stawarz, J. E., & Forman, M. A. 2012, ApJ, 754, 93
  * D’Amicis et al. (2010) D’Amicis, R., Bruno, R., Pallocchia, G., et al. 2010, ApJ, 717, 474
  * Di Mare et al. (2019) Di Mare, F., Sorriso-Valvo, L., Retinò, A., Malara, F., & Hasegawa, H. 2019, Atmosphere, 10, 561
  * Dobrowolny et al. (1980) Dobrowolny, M., Mangeney, A., & Veltri, P. 1980, Physical Review Letters, 45, 144
  * Dong et al. (2014) Dong, Y., Verdini, A., & Grappin, R. 2014, The Astrophysical Journal, 793, 118
  * Dudok De Wit (2004) Dudok De Wit, T. 2004, Phys. Rev. E, 70, 055302(R)
  * D’Amicis et al. (2021) D’Amicis, R., Perrone, D., Bruno, R., & Velli, M. 2021, Journal of Geophysical Research: Space Physics, 126, e2020JA028996, e2020JA028996 2020JA028996
  * Foldes et al. (2022) Foldes, R., Lévêque, E., Marino, R., et al. 2022, Journal of Plasma Physics (submitted)
  * Fraternale et al. (2016) Fraternale, F., Gallana, L., Iovieno, M., et al. 2016, Physica Scripta, 91, 023011
  * Frisch (1995) Frisch, U. 1995, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press)
  * Gogoberidze et al. (2013) Gogoberidze, G., Perri, S., & Carbone, V. 2013, ApJ, 769, 111
  * Grappin et al. (1991) Grappin, R., Velli, M., & Mangeney, A. 1991, Annales Geophysicae, 9, 416
  * Grappin et al. (1993) Grappin, R., Velli, M., & Mangeney, A. 1993, Phys. Rev. Lett., 70, 2190
  * Greco et al. (2012) Greco, A., Matthaeus, W. H., D’Amicis, R., Servidio, S., & Dmitruk, P. 2012, ApJ, 749, 105
  * Greco et al. (2016) Greco, A., Perri, S., Servidio, S., Yordanova, E., & Veltri, P. 2016, ApJ, 823, L39
  * Hadid et al. (2017) Hadid, L. Z., Sahraoui, F., & Galtier, S. 2017, ApJ, 838, 9
  * Heinemann & Olbert (1980) Heinemann, M. & Olbert, S. 1980, Journal of Geophysical Research: Space Physics, 85, 1311
  * Hernández et al. (2021) Hernández, C. S., Sorriso-Valvo, L., Bandyopadhyay, R., et al. 2021, ApJ, 922, L11
  * Hossain et al. (1995) Hossain, M., Gray, P. C., Pontius, Duane H., J., Matthaeus, W. H., & Oughton, S. 1995, Physics of Fluids, 7, 2886
  * Kasper et al. (2021) Kasper, J. C., Klein, K. G., Lichko, E., et al. 2021, Phys. Rev. Lett., 127, 255101
  * Kieokaew et al. (2021) Kieokaew, R., Lavraud, B., Yang, Y., et al. 2021, arXiv e-prints, arXiv:2103.15489
  * Kiyani et al. (2006) Kiyani, K., Chapman, S., & Hnat, B. 2006, Phys. Rev. E, 74, 051122
  * Kiyani et al. (2009) Kiyani, K. H., Chapman, S. C., & Watkins, N. W. 2009, Phys. Rev. E, 79, 036109
  * Kolmogorov (1941) Kolmogorov, A. N. 1941, Dokl. Akad. Nauk SSSR, 32, 16
  * Kraichnan (1965) Kraichnan, R. H. 1965, Phys. of Fluids, 8, 1385
  * Leamon et al. (1998) Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H., & Wong, H. K. 1998, J. Geophys. Res., 103, 4775
  * MacBride et al. (2005) MacBride, B. T., Forman, M. A., & Smith, C. W. 2005, in ESA Special Publication, Vol. 592, Solar Wind 11/SOHO 16, Connecting Sun and Heliosphere, ed. B. Fleck, T. H. Zurbuchen, & H. Lacoste, 613
  * Macek et al. (2012) Macek, W. M., Wawrzaszek, A., & Carbone, V. 2012, Journal of Geophysical Research (Space Physics), 117, 12101
  * Malara et al. (2000) Malara, F., Primavera, L., & Veltri, P. 2000, Physics of Plasmas, 7, 2866
  * Marino & Sorriso-Valvo (2023) Marino, R. & Sorriso-Valvo, L. 2023, Physics Reports, 1006, 1
  * Marino et al. (2008) Marino, R., Sorriso-Valvo, L., Carbone, V., et al. 2008, ApJ, 677, L71
  * Marino et al. (2011) Marino, R., Sorriso-Valvo, L., Carbone, V., et al. 2011, Planetary and Space Science, 59, 592
  * Marino et al. (2012) Marino, R., Sorriso-Valvo, L., D’Amicis, R., et al. 2012, ApJ, 750, 41
  * Matteini et al. (2018) Matteini, L., Stansby, D., Horbury, T. S., & Chen, C. H. K. 2018, ApJ, 869, L32
  * Matthaeus & Goldstein (1982) Matthaeus, W. H. & Goldstein, M. L. 1982, Journal of Geophysical Research: Space Physics, 87, 6011
  * Matthaeus & Goldstein (1986) Matthaeus, W. H. & Goldstein, M. L. 1986, Phys. Rev. Lett., 57, 495
  * Mininni et al. (2006) Mininni, P. D., Pouquet, A. G., & Montgomery, D. C. 2006, Phys. Rev. Lett., 97, 244503
  * Miura (2019) Miura, H. 2019, Fluids, 4
  * Montagud-Camps et al. (2018) Montagud-Camps, V., Grappin, R., & Verdini, A. 2018, ApJ, 853, 153
  * Montagud-Camps et al. (2020) Montagud-Camps, V., Grappin, R., & Verdini, A. 2020, ApJ, 902, 34
  * Montagud-Camps et al. (2022) Montagud-Camps, V., Hellinger, P., Verdini, A., et al. 2022, The Astrophysical Journal, 938, 90
  * Orszag & Tang (1979) Orszag, S. A. & Tang, C. M. 1979, Journal of Fluid Mechanics, 90, 129
  * Pagel & Balogh (2003) Pagel, C. & Balogh, A. 2003, J. Geophys. Res. (Space Physics), 108, 1012
  * Parashar et al. (2019) Parashar, T. N., Cuesta, M., & Matthaeus, W. H. 2019, ApJ, 884, L57
  * Parker (1965) Parker, E. N. 1965, Space Sci. Rev., 4, 666
  * Perrone et al. (2018) Perrone, D., Stansby, D., Horbury, T. S., & Matteini, L. 2018, Monthly Notices of the Royal Astronomical Society, 483, 3730
  * Politano & Pouquet (1998) Politano, H. & Pouquet, A. 1998, Geochim. Res. Lett., 25, 273
  * Quijia et al. (2021) Quijia, P., Fraternale, F., Stawarz, J. E., et al. 2021, MNRAS, 503, 4815
  * Roberts et al. (1992) Roberts, D. A., Goldstein, M. L., Matthaeus, W. H., & Ghosh, S. 1992, Journal of Geophysical Research: Space Physics, 97, 17115
  * Salem et al. (2009) Salem, C., Mangeney, A., Bale, S. D., & Veltri, P. 2009, ApJ, 702, 537
  * Smith et al. (2009) Smith, C. W., Stawarz, J. E., Vasquez, B. J., Forman, M. A., & MacBride, B. T. 2009, Phys. Rev. Lett., 103, 201101
  * Sorriso-Valvo et al. (1999) Sorriso-Valvo, L., Carbone, V., Veltri, P., Consolini, G., & Bruno, R. 1999, Geochim. Res. Lett., 26, 1801
  * Sorriso-Valvo et al. (2007) Sorriso-Valvo, L., Marino, R., Carbone, V., et al. 2007, Phys. Rev. Lett., 99, 115001
  * Sorriso-Valvo et al. (2015) Sorriso-Valvo, L., Marino, R., Lijoi, L., Perri, S., & Carbone, V. 2015, ApJ, 807, 86
  * Sorriso-Valvo et al. (2021) Sorriso-Valvo, L., Yordanova, E., Dimmock, A. P., & Telloni, D. 2021, ApJ, 919, L30
  * Stawarz et al. (2010) Stawarz, J. E., Smith, C. W., Vasquez, B. J., Forman, M. A., & MacBride, B. T. 2010, ApJ, 713, 920
  * Stawarz et al. (2011) Stawarz, J. E., Vasquez, B. J., Smith, C. W., Forman, M. A., & Klewicki, J. 2011, ApJ, 736, 44
  * Taylor (1938) Taylor, G. I. 1938, Royal Society of London Proceedings Series A, 164, 476
  * Telloni et al. (2021a) Telloni, D., Sorriso-Valvo, L., Woodham, L. D., et al. 2021a, ApJ, 912, L21
  * Telloni et al. (2021b) Telloni, D., Sorriso-Valvo, L., Woodham, L. D., et al. 2021b, ApJ, 912, L21
  * Telloni et al. (2022) Telloni, D., Zank, G. P., Stangalini, M., Downsal, C., & al. 2022, ApJ
  * Tenerani & Velli (2017) Tenerani, A. & Velli, M. 2017, ApJ, 843, 26
  * Totten et al. (1995) Totten, T. L., Freeman, J. W., & Arya, S. 1995, Journal of Geophysical Research: Space Physics, 100, 13
  * Tu & Marsch (1995) Tu, C. Y. & Marsch, E. 1995, Space Sci. Rev., 73, 1
  * Vasquez et al. (2007) Vasquez, B. J., Smith, C. W., Hamilton, K., MacBride, B. T., & Leamon, R. J. 2007, Journal of Geophysical Research (Space Physics), 112, A07101
  * Velli et al. (1990) Velli, M., Grappin, R., & Mangeney, A. 1990, Computer Physics Communications, 59, 153
  * Velli et al. (2020) Velli, M., Harra, L. K., Vourlidas, A., et al. 2020, A&A, 642, A4
  * Verdini & Grappin (2015) Verdini, A. & Grappin, R. 2015, ApJ, 808, L34
  * Verdini et al. (2015) Verdini, A., Grappin, R., Hellinger, P., Landi, S., & Müller, W. C. 2015, ApJ, 804, 119
  * Verdini et al. (2012) Verdini, A., Grappin, R., Pinto, R., & Velli, M. 2012, ApJ, 750, L33
  * Verma (2004) Verma, M. K. 2004, Phys. Reports, 401, 229
  * Verscharen et al. (2019) Verscharen, D., Klein, K. G., & Maruca, B. A. 2019, Living Reviews in Solar Physics, 16, 5
  * von Steiger et al. (2000) von Steiger, R., Schwadron, N. A., Fisk, L. A., et al. 2000, Journal of Geophysical Research: Space Physics, 105, 27217
  * Watson et al. (2022) Watson, A. S., Smith, C. W., Marchuk, A. V., et al. 2022, ApJ, 927, 43
  * Wu et al. (2022) Wu, H., Tu, C., He, J., Wang, X., & Yang, L. 2022, ApJ, 927, 113
  * Zhao et al. (2022) Zhao, L.-L., Zank, G. P., Telloni, D., et al. 2022, ApJ, 928, L15