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**Nonlinear Processes in Geophysics**
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NPG | Articles | Volume 25, issue 1

Nonlin. Processes Geophys., 25, 39–54, 2018

https://doi.org/10.5194/npg-25-39-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-25-39-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Nonlinear Waves and Chaos

**Review article**
30 Jan 2018

**Review article** | 30 Jan 2018

Intermittent turbulence in the heliosheath and the magnetosheath plasmas based on Voyager and THEMIS data

^{1}Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938 Warsaw, Poland^{2}Space Research Centre, Polish Academy of Science, Bartycka 18A, 00-716 Warsaw, Poland

^{1}Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938 Warsaw, Poland^{2}Space Research Centre, Polish Academy of Science, Bartycka 18A, 00-716 Warsaw, Poland

**Correspondence**: Wiesław M. Macek (macek@cbk.waw.pl)

**Correspondence**: Wiesław M. Macek (macek@cbk.waw.pl)

Abstract

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Turbulence is complex behavior that is ubiquitous in space, including the environments of the heliosphere and the magnetosphere. Our studies on solar wind turbulence including the heliosheath, and even at the heliospheric boundaries, also beyond the ecliptic plane, have shown that turbulence is intermittent in the entire heliosphere. As is known, turbulence in space plasmas often exhibits substantial deviations from normal Gaussian distributions. Therefore, we analyze the fluctuations of plasma and magnetic field parameters also in the magnetosheath behind the Earth's bow shock. Based on THEMIS observations, we have already suggested that turbulence behind the quasi-perpendicular shock is more intermittent with larger kurtosis than that behind the quasi-parallel shocks. Following this study, we would like to present a detailed analysis of intermittent anisotropic turbulence in the magnetosheath depending on various characteristics of plasma behind the bow shock and now also near the magnetopause. In particular, for very high Alfvénic Mach numbers and high plasma beta we have clear non-Gaussian statistics in the directions perpendicular to the magnetic field. On the other hand, for directions parallel to this field the kurtosis is small and the plasma is close to equilibrium. However, the level of intermittency for the outgoing fluctuations seems to be similar to that for the ingoing fluctuations, which is consistent with approximate equipartition of energy between the oppositely propagating Alfvén waves. We hope that the difference in characteristic behavior of these fluctuations in various regions of space plasmas can help to detect some complex structures in space missions in the near future.

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Macek, W. M., Wawrzaszek, A., and Kucharuk, B.: Intermittent turbulence in the heliosheath and the magnetosheath plasmas based on Voyager and THEMIS data, Nonlin. Processes Geophys., 25, 39–54, https://doi.org/10.5194/npg-25-39-2018, 2018.

1 Introduction

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Turbulence is complex behavior that is ubiquitous in space, including the solar wind, interplanetary and interstellar media, as well as planetary and interstellar shocks (e.g., Bruno and Carbone, 2016). These shocks are usually collisionless and processes responsible for the plasma are substantially different from ordinary gases; see, e.g., Kivelson and Russell (1995) and Burgess and Scholer (2015). That is, the necessary coupling in plasma is usually provided by nonlinear structures at various scales, possibly exhibiting fractal or multifractal self-similarity properties (e.g., Burlaga, 1995; Macek, 2006). In addition, dissipation (so-called quasi-viscosity) could often result from wave damping or other processes related to electric current structures. The mechanism of complexity of space and astrophysical plasmas is still a challenge to turbulence problems (Chang, 2015).

2 Multifractal model

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In our view, we should still rely on phenomenological models of
intermittent turbulence, which can grasp multiplicative processes
leading to complex behavior of the plasma in a simple way. As we
have often argued (e.g., Macek, 2006, 2007; Macek and Wawrzaszek, 2009), the most
useful concept for such a phenomenological study is a topological
object, namely the generalized two-scale weighted Cantor set – an
example of multifractals – as described, for example, by
Falconer (1990). The turbulence model based on this set is sketched
here in Fig. 1, as taken from Macek (2007). We see that at each
step of construction of the generalized Cantor set, one needs to
specify two scales *l*_{1} and *l*_{2} (${l}_{\mathrm{1}}+{l}_{\mathrm{2}}\le \mathrm{1}$)
associated with probability measures *p* and 1−*p*. In fact,
fractals and multifractals could be considered a convenient
mathematical language useful for understanding dynamics of
turbulence, as already postulated by Mandelbrot (1982). In fact, in
this review we will provide some arguments that this surprisingly
simple mathematical rule provides a very efficient tool for
phenomenological analysis of complex turbulent media.

Moreover, for the two-scale weighted Cantor set model, the singularity multifractal spectrum shown in Fig. 2 can easily be calculated (e.g., Ott, 1993). In particular, the width of this universal function, Δ, is obtained analytically by the following equation:

$$\begin{array}{}\text{(1)}& {\displaystyle}{\displaystyle}\mathrm{\Delta}\equiv {\mathit{\alpha}}_{\text{max}}-{\mathit{\alpha}}_{\text{min}}={D}_{-\mathrm{\infty}}-{D}_{\mathrm{\infty}}=\left|{\displaystyle \frac{\mathrm{log}(\mathrm{1}-p)}{\mathrm{log}{l}_{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{log}\left(p\right)}{\mathrm{log}{l}_{\mathrm{1}}}}\right|.\end{array}$$

Naturally, this quantity Δ is just the difference between the maximum and minimum dimensions related to the regions in the phase space with the least dense and most dense probability densities, and hence it has been proposed by Macek (2007) and Macek and Wawrzaszek (2009) as a degree of multifractality. Moreover, since this parameter Δ exhibits a deviation from a strict self-similarity, it can also be used as a degree of intermittency, as explained in Frisch (1995, Chapter 8). One can expect that the solar wind Δ will reveal various nonlinear phenomena, including nonlinear pressure pulses related to magnetosonic waves, as argued by Burlaga et al. (2003, 2007).

The other parameter *A* describing the multifractal scaling is the measure of
asymmetry of the spectrum as defined by Macek and Wawrzaszek (2009):

$$\begin{array}{}\text{(2)}& {\displaystyle}{\displaystyle}A\equiv {\displaystyle \frac{{\mathit{\alpha}}_{\mathrm{0}}-{\mathit{\alpha}}_{\text{min}}}{{\mathit{\alpha}}_{\text{max}}-{\mathit{\alpha}}_{\mathrm{0}}}},\end{array}$$

where *α*=*α*_{0} is the point at which the spectrum has
its maximum, *f*(*α*_{0})=1. In particular, in a simpler case
when *A*=1 (${l}_{\mathrm{1}}={l}_{\mathrm{2}}=\mathrm{0.5}$), the one-scale *p* model is recovered
(e.g., Meneveau and Sreenivasan, 1987), and for a monofractal the function in
Fig. 2 is reduced to a point.

In principle, for experimental time series one can recover the
multifractal spectrum and fit to either the well-known *p* model or the
more general two-scale weighted Cantor set model. For Voyager data
this can be done in the following way. That is, the generalized
multifractal measures *p*(*l*) depending on scale *l* can be
constructed using magnetic field strength fluctuations
(Burlaga, 1995). Normalizing a time series of daily averages
*B*(*t*_{i}), where $i=\mathrm{1},\mathrm{\dots},N={\mathrm{2}}^{n}$ for $j={\mathrm{2}}^{n-k}$, $k=\mathrm{0},\mathrm{1},\mathrm{\dots},n$,

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}p({x}_{j},l)\equiv {\displaystyle \frac{\mathrm{1}}{N}}\sum _{i=\mathrm{1}+(j-\mathrm{1})\mathrm{\Delta}t}^{j\mathrm{\Delta}t}B\left({t}_{i}\right)={p}_{j}\left(l\right)\end{array}$$

is calculated with the successive average values 〈*B*(*t*_{i},Δ*t*)〉 of *B*(*t*_{i}) between *t*_{i} and *t*_{i}+Δ*t*,
for each Δ*t*=2^{k} (Macek et al., 2011, 2012). When time
series are obtained onboard spacecraft, it is usually possible to
relate time dependence to space dependence by using the Taylor (1938)
hypothesis. Because the average solar wind speed *v*_{sw}
is much greater that the velocity of the space probe, we can argue
that *p*(*x*_{j},*l*) can be regarded as the probability that at
a position *x*=*v*_{sw}*t*, at time *t*, a given magnetic
flux will be transferred to a spatial scale *l*=*v*_{sw}Δ*t*.

In this way Burlaga (1995) has shown that in the inertial range the
average value of the *q*th moment of *B* at various scales *l*
scales as

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle}\langle {B}^{q}\left(l\right)\rangle \sim {l}^{\mathit{\gamma}\left(q\right)},\end{array}$$

where the exponent *γ* is related to the generalized
dimension, $\mathit{\gamma}\left(q\right)=(q-\mathrm{1})({D}_{q}-\mathrm{1})$. If for a certain range
of spatial scales *l* corresponding to a given interval of time
Δ*t* we have a straight line on a logarithmic scale using
these slopes for each real *q*, the values of *D*_{q} can be
determined with Eq. (4).

Alternatively, as explained by Macek and Wawrzaszek (2009), the multifractal
function *f*(*α*) vs. scaling index *α* shown in
Fig. 2, which exhibits universality of the
multifractal scaling behavior, can be obtained using the Legendre
transformation. It is worth noting, however, that we obtain this
multifractal universal function directly from the slopes for
a given scale range using this direct method in various situations
(see Macek and Wawrzaszek, 2009; Macek et al., 2011, 2012, 2014).

3 Heliosheath turbulence

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The schematic of the heliospheric boundaries is shown in Fig. 3. Voyager 1 entered the heliosheath after crossing the termination heliospheric shock at 94 AU in 2004, while Voyager 2 crossed this shock at 84 AU in 2007. It is generally accepted that after crossing the heliopause in 2012, the last boundary separating the heliosphere from the nearby interstellar medium, the Voyager 1 has ultimately left the heliosphere, while the crossing of the heliopause by Voyager 2 is expected in the very near future.

The main aim of our Voyager studies is to look at the measure of multifractal scaling in the heliosheath. Because in the distant heliosphere the magnetic fields have mainly azimuthal components, one can use the magnitude of the magnetic fields $\left|\mathit{B}\right|$ to estimate the probability measures and using straight lines according to Eq. (4) in a certain scale range, as with those seen in Figs. 1 and 2 of the paper by Macek et al. (2014), and in this way we can calculate the multifractal singularity spectrum. The results using the data gathered onboard both the Voyager 1 and 2 spacecraft immersed in the heliosheath are presented in Fig. 4, case (a) at 94–97 AU for the year 2005 and (c) at 105–107 AU for the year 2008 for Voyager 1, and Voyager 2 in case (b) at 85–88 AU for the year 2008 and (d) at 88–90 AU for the year 2009, respectively (Macek et al., 2012, Fig. 5).

It seems that the two-scale weighted Cantor set model fits the
data better than the classical *p* model. To support this result
in a more quantitative way, we have used the weighted *χ*^{2},
consisting of a sum of squares of differences between the spectrum
obtained from data and the model, each normalized to unit variance
(e.g., Press et al., 1992). This measure of fit quality in the two
cases is shown in Table 1. We see that the values obtained for
the two-scale model are at least one order in magnitude smaller than
that for the standard *p* model. This means that the generalized
Cantor set model is in fact substantially better.

We have also calculated the degree of multifractality Δ, as
given in Eq. (1), for Voyager 1 in the heliosheath
depending on the heliospheric distances during different phases of
the solar cycle, the minimum (MIN), maximum (MAX), declining (DEC), and
rising (RIS) phases, which is now demonstrated in
Fig. 5 (left panel). We clearly see that this
multifractality measure obtained for the scale range of Δ*t*
from 2 to 16 days decreases steadily with the heliospheric
distance and is modulated by the solar activity following the
sunspot numbers (SSN, indicated at the bottom panel), as taken
from Macek et al. (2011, Fig. 2). In the right panel we show Δ
calculated in the heliosheath for two various scaling ranges from
2 to 16 and from 4 to 32 days (cf. Macek et al., 2014). The crossings of
the termination shock (TS) and the heliopause (HP) by Voyager 1
are indicated by vertical dashed lines. We see that in the
heliosheath the degree of multifractality basically still follows
the periodic dependence fitted inside the heliosphere
(Macek et al., 2011, 2012). It is worth noting that after crossing
the heliopause at ∼122 AU, the value of Δ suddenly
drops to zero, and nonmultifractal (nonintermittent) smoothly
varying magnetic fields are observed by Voyager 1, as indicated in
the right panel of Fig. 5; see Macek et al. (2014).
This would mean that the entire heliosphere with turbulent plasma
inside is immersed in a relatively quiet ambient very local
interstellar medium.

Naturally, the multifractal spectrum can be related to nonlinear Alfvén waves, associated with discontinuities, or mirror mode structures due to some plasma instabilities, or possibly current sheets (Borovsky, 2010; Tsurutani et al., 2011a, b) generated upstream of the termination shock, as discussed in our previous paper (Macek and Wawrzaszek, 2013). In this way we have applied the multifractal model (Macek, 2007; Macek and Szczepaniak, 2008) to solar wind turbulence in the entire heliosphere (Szczepaniak and Macek, 2008; Macek and Wawrzaszek, 2009; Macek et al., 2011, 2012), also beyond the ecliptic plane (Wawrzaszek and Macek, 2010; Wawrzaszek et al., 2015), and even at the heliospheric boundaries (Burlaga et al., 2013; Macek et al., 2014), and have shown that turbulence could often be intermittent. By the way, it would be difficult to argue that there is an asymmetry in these spectra for the Voyager 1 data, but there are some deviations from the symmetric spectrum for Voyager 2. In summary, the values of the degree of intermittency calculated from our two-scale weighted Cantor set model are presented in Fig. 6 (cf. Macek, 2012).

As is known, turbulence in space and astrophysical plasmas exhibits deviations from normal distributions, and these higher moments are often considered signatures of intermittency. In particular, kurtosis – the fourth moment of the probability density function – is often used as a measure of intermittency (Bruno et al., 2003; Bruno and Carbone, 2013).

4 Magnetosheath turbulence

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Naturally, nonlinear structures responsible for turbulence have already been identified in planetary environments, in the solar wind, and also in the magnetosheath (e.g., Alexandrova, 2008). In particular, the magnetic fluctuations using Wind (Lion et al., 2016) and Cluster multi-spacecraft have been analyzed at ion scales (Yordanova et al., 2008; Roberts et al., 2016; Perrone et al., 2016, 2017). In addition, some results on very high-resolution data on electron scales have recently been provided by the Magnetospheric Multiscale (MMS) mission (Yordanova et al., 2016; Chasapis et al., 2017). Moreover, on the basis of kinetic simulations by Karimabadi et al. (2014), one can suggest some interesting relationships of turbulent processes near shocks with reconnection processes. But in spite of progress in MHD simulations, including Hall effects, the physical mechanisms of turbulent behavior are still not sufficiently clear.

Various space missions provide unique observational data, which help to understand phenomena in our environment in space. In particular, the THEMIS mission was launched by NASA in 2007 in order to resolve macroscale phenomena occurring during substorms (Sibeck and Angelopoulos, 2008), as schematically presented in Fig. 7. In addition, for the first time THEMIS data were used for analysis of turbulence at the terrestrial bow shock. That is, we have suggested that turbulence behind the quasi-perpendicular shock is more intermittent with larger kurtosis than that behind the quasi-parallel shocks (Macek et al., 2015).

In this review paper, besides turbulence in the heliosheath, as has already been discussed in Sect. 3, now in Sect. 4 we continue our study in the entire magnetosheath also near the magnetopause. However, since it would be difficult to obtain the full multifractal spectrum using the THEMIS data, at present we only examine how the degree of multifractality resulting in deviation from the normal distribution, which is also a level of intermittency, depends on the characteristics of the solar wind and magnetospheric plasmas (Macek et al., 2017). The data under study are briefly described in Sect. 4.1. In Sect. 4.2 we present the results of our analysis, showing in particular that at high Alfvénic Mach numbers turbulence becomes clearly intermittent. The importance of this intermittent behavior for space plasmas is underlined in Sect. 5.

We analyze various time samples acquired during the long period between 2008 and 2015 from the THEMIS mission consisting of a quintet (A, B, C, D, and E) of space probes (Sibeck and Angelopoulos, 2008), as listed in Table 2. We have selected the following 24 intervals in the magnetosheath (without any evident large-scale static plasma structures): 11 samples measured after crossing the bow shock, denoted by BS, and 13 samples obtained before leaving the magnetosheath, i.e., near the magnetopause, denoted by MP. The time resolution here is 3 s and these samples taken in the Geocentric Solar Ecliptic (GSE) reference system are all (except for no. 11) longer than 4 h. Naturally, the length of each sample depends on the orbit of a particular probe immersed in the magnetosheath during some periods of time. Please note that the timescales in the magnetosheath are much shorter than that in the heliosheath.

Various characteristic plasma parameters, namely the Alfvén
Mach number, *M*_{A}, the plasma parameter beta, *β*,
and the magnetosonic Mach number, *M*_{ms}, are calculated
in the solar wind upstream: first before crossing the bow shock
(before entering the magnetosheath) and next in the magnetosphere
(before crossing the magnetopause). The plasma *β* is the
ratio of the thermal pressure *p* to the magnetic pressure
*B*^{2}∕(2*μ*_{0}*ρ*), where *ρ*=*m**N* is the mass
density for ions of mass *m* and the number density *N* (*μ*_{0}
denotes the permeability of free space).

All three of these plasma parameters vs. sample number are depicted in
Fig. 8. We see that the Alfvén Mach
numbers can vary substantially with the limiting value of about 25
($\mathrm{5}\le {M}_{\mathrm{A}}\le \mathrm{25}$), and that in most cases
*β* is below 5 (only three cases are above 10). However, the magnetosonic
Mach numbers are rather moderate: $\mathrm{3.6}\le {M}_{\text{ms}}\le \mathrm{7.5}$.

Using the values of plasma and magnetic fields shown in Figs. 1 and
2 of the paper by Macek et al. (2017), we can calculate the
Elsässer variables, ${\mathit{z}}^{\pm}=\mathit{V}\pm {\mathit{V}}_{\mathrm{A}}$, where the characteristic Alfvénic
velocity is given by ${\mathit{V}}_{\mathrm{A}}=\mathit{B}/({\mathit{\mu}}_{\mathrm{0}}\mathit{\rho}{)}^{\mathrm{1}/\mathrm{2}}$ (Elsasser, 1950). It is worth noting that the sign is taken
here relative to the local average magnetic field *B*_{o},
which certainly depends on the timescale *τ* responsible for turbulence
(Kiyani et al., 2013), as recently noted by Gerick et al. (2017).
Because the time period during which this average background
magnetic field is calculated, say *d**τ*, should be substantially
larger than the timescale of turbulence *τ*, we have taken *d*=10.
By the way, in turbulence the dependence of statistical moments on
spatial scales is often considered. For example, based on
spacecraft measurements in the solar wind, one can estimate spatial
scales by using the Taylor (1938) hypothesis
(e.g., Macek and Wawrzaszek, 2009). However, in the magnetosheath the solar
wind velocity is substantially reduced, and this approach can be
somewhat less certain (Mangeney et al., 2006), especially for some plasma
parameters (Perri et al., 2017). Therefore, it is better to analyze
directly time samples obtained onboard several space probes, as is
the case with the THEMIS mission.

Now, following our previous work on THEMIS data, the kurtosis of
the increments of the various components of both Elsässer
vectors *z*^{±}, $\mathit{\delta}{\mathit{z}}^{\pm}(t,\mathit{\tau})={\mathit{z}}^{\pm}(t+\mathit{\tau})-{\mathit{z}}^{\pm}\left(t\right)$, can be calculated
for any given scale *τ*, taken in units of time resolution
(Macek et al., 2015, Eq. 1). As is known, the Alfvénic increments
perpendicular to the direction of *B*_{o} and the parallel
compressive (slow-mode-like) increments should provide rather
different contributions to the turbulent behavior of the solar wind
plasma (e.g., Bruno et al., 2003; Oughton and Matthaeus, 2005). Therefore, we have
performed our calculation in the Mean Field (MF) coordinate system,
as described by Bruno and Carbone (2013). That is, the direction parallel
to the local mean field *B*_{o} in the GSE system is taken along the versor
$\widehat{\mathit{z}}={\mathit{B}}_{\mathrm{o}}/{B}_{o}$ of the new MF reference system (the symbol
$\widehat{\phantom{\rule{0.25em}{0ex}}}$ is used for a unitary vector). This allows us to calculate
the parallel components of both Elsässer vectors $\mathit{\delta}{z}_{\parallel}^{+}$ and $\mathit{\delta}{z}_{\parallel}^{-}$. Next, in order to
obtain two other components perpendicular to the field
*B*_{o}, $\mathit{\delta}{z}_{\u27c2\mathrm{1}}^{+}$, $\mathit{\delta}{z}_{\u27c2\mathrm{1}}^{-}$ and
$\mathit{\delta}{z}_{\u27c2\mathrm{2}}^{+}$, $\mathit{\delta}{z}_{\u27c2\mathrm{2}}^{-}$, we take for the
latter case the axis in the direction perpendicular to the plane
containing the mean field *B*_{o} and the
** X** axis
in the GSE system (taken here as positive from the Sun), which is
approximately consistent with the radial component of the mean
solar wind velocity (

The obtained values of kurtosis of the increments of the
fluctuations of the Elsässer variables for the outgoing and
ingoing Alfvénic fluctuations, respectively,
*z*^{+} and
*z*^{−}, as observed by THEMIS in the magnetosheath near the
bow shock (BS, red circles) and magnetopause (MP, white triangles) vs. the Alfvén Mach number, the total plasma beta
*β*, and the magnetosonic Mach number, corresponding to
Fig. 8, are presented in
Figs. 9–11 for all 24 cases
listed in Table 2.
The departure of the probability density functions from normal
distributions for the selected four cases corresponding to Figs. 1
and 2 of the paper by Macek et al. (2017), namely near the bow
shock, cases (a) and (b), and near the magnetopause, cases (c) and
(d), for a given timescale *τ* = 9 s, are illustrated
in Figs. 12 and 13, respectively.
The dependence of the kurtosis on the timescale *τ* is
depicted in the corresponding Figs. 14 and
15.

Figures 9–11 show kurtosis for the increments of all the
components of the Elsässer vectors, for all the cases listed in
Table 2, but for only one scale. Even though there is no very
clear dependence on these plasma parameters, one can notice that the
value of kurtosis often decreases with Alfvénic Mach number
along the local magnetic field and sometimes increases in the
perpendicular directions. We see from Fig. 9
that near the bow shock for the outgoing fluctuations kurtosis
along the magnetic field $\mathit{\delta}{z}_{\parallel}^{+}$ somewhat
decreases from 16.56 ± 0.06 at lower *M*_{A} (bin: $\mathrm{5}\le {M}_{\mathrm{A}}\le \mathrm{15}$) to 10.28 ± 0.06 at higher
*M*_{A} (bin: $\mathrm{15}<{M}_{\mathrm{A}}\le \mathrm{25}$, even though we have only
two points in this bin). But near the magnetopause, where
we have more points in the letter bin, we can observe a clear
significant decrease from 12.58 ± 0.05 to 7.25 ± 0.05. We have
basically a similar behavior for the ingoing fluctuations, $\mathit{\delta}{z}_{\parallel}^{-}$: a decrease from 19.69 ± 0.06 to
16.86 ± 0.06 at the BS and a clear decrease from 17.66 ± 0.05 to
6.20 ± 0.05 at the MP. Here the standard deviations of kurtosis are
rather small, about 0.06, as calculated according to
Press et al. (1992). However, for the transverse components $\mathit{\delta}{z}_{\u27c2\mathrm{1},\mathrm{2}}^{\pm}$, kurtosis seems to be more scattered and often rather
increasing with the Alfvénic Mach number, seeming not only anisotropic,
but also seeming to be non-gyrotropic, with
differences in two perpendicular components. We can see from
Fig. 10 that $\mathit{\delta}{z}_{\parallel}^{\pm}$
decrease with plasma *β*, approaching normal distribution for
high *β*, when the thermal pressure dominates the plasma
behavior. But we do not see any clear regularity for the
dependence of $\mathit{\delta}{z}_{\u27c2\mathrm{1},\mathrm{2}}^{\pm}$ on *β*. It also
seems from Fig. 11 that the value of kurtosis
is not very sensitive to the magnetosonic Mach number, but
admittedly the range of this parameter considered in Table 2 is rather
limited: $\mathrm{3.6}\le {M}_{\text{ms}}\le \mathrm{7.5}$.

Additionally, for the four clearly quasi-perpendicular cases
(illustrated in Figs. 1 and 2 of the work by
Macek et al., 2017, cases 3, 5, 15, and 16 listed in Table 2),
the dependence of the parallel and perpendicular components of
kurtosis on timescale *τ* for both the outgoing (*z*^{+})
and ingoing fluctuations (*z*^{−}) is now presented in
Figs. 14 and 15, taken from
(Macek et al., 2017). We can see that kurtosis behind the bow
shock, Fig. 14, cases (a) and (b) in Table 2,
could sometimes (for *β*∼1) be smaller than that near the
magnetopause, Fig. 15, cases (c) and (d) in
Table 2. We can generally notice only small differences between
*z*^{+} and *z*^{−}, and therefore the outgoing and
ingoing fluctuations seem to be similar, which is
roughly consistent with
equipartition suggested by Tu et al. (1989). On the other hand,
behind the bow shock for small plasma *β*∼1 (when the
thermal pressure and the magnetic pressure are similar in the
magnetized plasma), but with a moderate Alfvénic Mach number
*M*_{A}≈9, Fig. 14a' and a”,
case (a) in Table 2, we see only small kurtosis with approximately
Gaussian normal distribution (i.e., close to equilibrium). For
similar *M*_{A}≈12 and somewhat higher plasma
*β*∼4, Fig. 14b' and b”, case (b) in
Table 2, both parallel and perpendicular components of the
Elsässer vectors are active. A similar behavior is also
observed near the magnetopause, Fig. 15c' and c”, case (c) in Table 2. Finally, it is worth noting that for the
highest value of *M*_{A} = 25 and *β* = 16.5, as
illustrated in Fig. 15d' and d” (case (d) in
Table 2), the perpendicular $\mathit{\delta}{z}_{\u27c2\mathrm{1},\mathrm{2}}^{\pm}$
components of fluctuations of Elsässer vectors are much larger
than the parallel $\mathit{\delta}{z}_{\parallel}^{\pm}$ components. This
exhibits a clear intermittent anisotropic turbulence with
non-Gaussian probability distributions in transverse directions.
On the other hand, the plasma along the local magnetic field is
rather close to equilibrium.

Even though there is no clear regularity in
Figs. 14 and 15 showing
dependence of the kurtosis on scale *τ*, it seems that kurtosis
near the bow shock (Fig. 14) is rather similar
to that near the magnetopause (Fig. 15).
That is, based on Figs. 9 and 10,
it seems that near the bow shock (circles)
the intermittency seems to decrease with the Alfvénic Mach
number *M*_{A} and decrease with the plasma beta *β*
near the magnetopause (triangles). We also see some difference
between *z*^{+} and *z*^{−} in Fig. 14b' and b”,
behind the bow shock (BS), and some scatters near the magnetopause (MP) for small scales
(Fig. 15c”, for *τ*<30 s), and therefore we
could consider the
other cases in Table 2. In fact, generally speaking, we have
verified that the level of intermittency for the outgoing
fluctuations *z*^{+} is usually similar to that for the ingoing
fluctuations *z*^{−}, which exhibits approximate equipartition
of energy between these oppositely propagating Alfvén waves.

5 Conclusions

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Using our weighted two-scale Cantor set model, which is
a convenient tool to investigate the asymmetry of the multifractal
spectrum, we confirm the characteristic shape of the universal
multifractal singularity spectrum. In fact, as seen in
Fig. 4, *f*(*α*) is a downward concave function
of scaling indices *α*. We show that the degree of
multifractality for magnetic field fluctuations of the solar wind
falls steadily with the distance from the Sun and seems to be
modulated by the solar activity also in the heliosheath. Moreover,
we have considered the multifractal spectra of fluctuations of the
interplanetary magnetic field strength before and after crossing of
the heliospheric termination shock by Voyager 1 and 2 near 94 and
84 AU from the Sun, respectively.

Further, we have provided important evidence that the large-scale magnetic field fluctuations reveal the multifractal structure not only in the outer heliosphere, but also in the entire heliosheath, even near the heliopause. Naturally, the evolution of the multifractal distributions should be related to some physical (MHD) models, as suggested by Burlaga et al. (2003, 2007). The driver of the multifractality in the heliosheath could be the solar variability on scales from hours to days, fast and slow streams or shock interactions, and other nonlinear structures discussed by Macek and Wawrzaszek (2013). In our view, any accurate physical model must reproduce the multifractal spectra. In particular, the observed nonmultifractal scaling after the heliopause crossing suggests nonintermittent behavior in the nearby interstellar medium, consistent with the smoothly varying interstellar magnetic field reported by Burlaga and Ness (2014). We have identified the scaling region of fluctuations of the interplanetary magnetic field.

In fact, using our two-scale model based on the weighted Cantor set, we have examined the universal multifractal spectra before and after crossing by Voyager 1: the termination shock at 94 AU and before crossing the heliopause at distances of about 122 AU from the Sun. Moreover, inside the heliosphere we observe the asymmetric spectrum, which becomes more symmetric in the heliosheath. We confirm that multifractality of magnetic field fluctuations embedded in the solar wind plasma for large scales decreases slowly with the heliospheric distance, demonstrating that this quantity is still modulated by the solar cycles further in the heliosheath, and even in the vicinity of the heliopause, possibly approaching a uniform nonintermittent behavior in the nearby interstellar medium. We propose this change in behavior as a signature of the expected crossing of the heliopause by Voyager 2 in the near future.

Regarding the magnetosheath, we have shown that turbulence for small
scales is intermittent in the entire magnetosheath, in regions near
the bow shock, and even near the magnetopause. In particular, we
have found that near the magnetopause at very high Alfvénic
Mach numbers *M*_{A} and high plasma *β* the
probability density functions of compressive fluctuations parallel
to the local average magnetic field should be nearly normal and
close to equilibrium with small kurtosis, while in the transverse
Alfvénic turbulence, resulting from nonlinear interactions, is
non-gyrotropic with large kurtosis for the Elsässer variables.
These fluctuations are more intermittent than that at the lower
Alfvénic Mach numbers and plasma beta behind the bow shock. On
the other hand, the level of intermittency for the outgoing
fluctuations (*z*^{+}) seems to be approximately similar to that for
the ingoing fluctuations (*z*^{−}). In view of the space
investigation in the near future, including the THOR mission
(e.g., Vaivads et al., 2016), we expect that the difference in
characteristic behavior of these fluctuations in various regions of
the magnetosheath will be able to help in identifying some new complex
structures
in space plasmas.

Data availability

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Data availability.

THEMIS mission data are available online from http://cdaweb.gsfc.nasa.gov (GSFC, 2018).

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Nonlinear Waves and Chaos”. It is a result of the 10th International Nonlinear Wave and Chaos Workshop (NWCW17), San Diego, United States, 20–24 March 2017.

Acknowledgements

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Acknowledgements.

We would like to thank the magnetic field instruments team of the Voyager
mission, the NASA National Space Science Data Center, and the Space
Science Data Facility for providing Voyager data. The research
leading to these results received funding from the THEMIS project during
a visit by WMM to the NASA Goddard Space Flight Center.
We would like to thank the plasma and magnetic field instruments team
of the THEMIS mission for providing the data, which are available online
from http://cdaweb.gsfc.nasa.gov. This work has been supported
by the National Science Center, Poland (NCN), through grant
2014/15/B/ST9/04782.

Edited by: George Morales

Reviewed by: three anonymous referees

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Special issue

Short summary

The aim of this review is a comparison of intermittency in the heliosphere and the magnetosphere. We show that turbulence in the heliosheath and even at the heliospheric boundaries is intermittent. Moreover, analysis of turbulence in the magnetosheath shows that for very high Alfvénic Mach numbers and high plasma beta we have non-Gaussian statistics in the transverse directions. However, for directions parallel to the magnetic fields, the plasma is close to equilibrium.

The aim of this review is a comparison of intermittency in the heliosphere and the...

Nonlinear Processes in Geophysics

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