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.

Turbulence is complex behavior that is ubiquitous in space,
including the solar wind, interplanetary and interstellar media,
as well as planetary and interstellar shocks

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

Two-scale weighted Cantor set model for asymmetric solar wind turbulence

The singularity multifractal spectrum

Moreover, for the two-scale weighted Cantor set model, the
singularity multifractal spectrum shown in Fig.

Naturally, this quantity

The other parameter

In principle, for experimental time series one can recover the
multifractal spectrum and fit to either the well-known

In this way

Alternatively, as explained by

The schematic of the heliospheric boundaries is shown in
Fig.

Schematic of the heliospheric boundaries (credit: NASA/Walt Feimer).

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

The singularity spectrum

The measure of chi-square fitting for the weighted two-scale and one-scale

The parameter

The degree of multifractality

It seems that the two-scale weighted Cantor set model fits the
data better than the classical

We have also calculated the degree of multifractality

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

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

Schematic of the THEMIS mission (credit: NASA/ESA).

Naturally, nonlinear structures responsible for turbulence have
already been identified in planetary environments, in the solar wind, and
also in the magnetosheath

List of selected interval samples (mm.dd.hh.MM).

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

In this review paper, besides turbulence in the heliosheath, as has
already been discussed in Sect.

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

Various characteristic plasma parameters, namely the Alfvén
Mach number,

Alfvén Mach number

All three of these plasma parameters vs. sample number are depicted in
Fig.

Kurtosis of the increments of the Elsässer vectors,

Kurtosis of the increments of the Elsässer vectors,

Kurtosis of the increments of the Elsässer vectors,

Using the values of plasma and magnetic fields shown in Figs. 1 and
2 of the paper by

Now, following our previous work on THEMIS data, the kurtosis of
the increments of the various components of both Elsässer
vectors

The obtained values of kurtosis of the increments of the
fluctuations of the Elsässer variables for the outgoing and
ingoing Alfvénic fluctuations, respectively,

The probability density functions (PDFs) of the increments of the parallel
(white circles) and two perpendicular (black diamonds and squares) components
of the Elsässer variables,

The probability density functions (PDFs) of the increments of the parallel
(white circles) and perpendicular (black diamonds and squares) components of
the Elsässer variables,

Kurtosis for the increments of the parallel (white circles) and two
perpendicular (black diamonds and squares) components of the Elsässer
variables,

Kurtosis for the increments of the parallel (white circles) and perpendicular
(black diamonds and squares) components of the Elsässer variables,

Figures

Additionally, for the four clearly quasi-perpendicular cases
(illustrated in Figs. 1 and 2 of the work by

Even though there is no clear regularity in
Figs.

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.

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

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

THEMIS mission data are available online from

The authors declare that they have no conflict of interest.

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.

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