A method is proposed to experimentally determine the intrinsic timescale or a decay rate of turbulent fluctuations. The method is based on the assumption that the Breit–Wigner spectrum model with a non-Gaussian frequency broadening is valid in the data analysis. The decay rate estimate is applied to the multispacecraft magnetic field data in interplanetary space, yielding the decay rate on spatial scales of about 1000 km (about 10 times larger than the ion inertial length), which is higher than the theoretical predictions from the random sweeping timescale of the eddy turnover time. The faster decay of fluctuation components in interplanetary space is interpreted as a realization of plasma physical (and not fluid mechanical) processes.

Turbulent fluctuations appear in various fluid or gaseous media and exhibit both temporally developing and spatially varying fields. The fluctuation properties are characterized by correlations between two spatial points and between two different times. A Fourier representation of the two-time, two-point correlation is the energy spectrum in the wavenumber–frequency domain.

Our experimental access to the wavenumber–frequency spectrum is limited. One
can easily obtain the frequency spectra from time series data using a single
probe, whereas one has to determine the wavenumber spectra using multiple
probes simultaneously. Moreover, the Fourier transform cannot be performed if
the number of probes is small (say, below 10). For this reason, Taylor's
frozen-in flow hypothesis

Time evolution is neglected during the time period of measurements when one uses Taylor's hypothesis, so the intrinsic timescale information cannot be estimated from single-point measurements. Of course, Taylor's hypothesis breaks down whenever the frozen-in condition of the fluctuating field is violated, for example, when the flow speed has large-scale variations (the random sweeping model), or when propagating waves exist in the turbulent fluctuations. Is it then ever possible to experimentally determine the intrinsic timescale in turbulence studies?

Here we develop a method to experimentally determine the lifetime of
fluctuation components in plasma turbulence. The task of measuring the
wavenumber–frequency spectrum is achieved by using multipoint magnetic field
measurements by Cluster spacecraft in interplanetary space

It is worthwhile to note that other choices are possible for analytically expressing or fitting the non-Gaussian shape of the frequency-sliced spectrum (a slice of the energy spectrum over the frequencies at a fixed value of the wavenumber). The kappa distribution is a likely candidate. Underlying physical models are different depending on whether the kappa distribution or the Breit–Wigner distribution is used. The use of kappa distribution is developed for describing non-extensive statistical mechanics (e.g., non-extensive entropy) and is developed for discrete particles that have long-range interactions and correlations. Its application to the energy spectra for continuous turbulent fields is still in question, in particular, in interpreting the control parameter kappa in the distribution . However, the use of a Breit–Wigner (or Lorentzian) distribution has a solid background with a physical model in that the decay rate (appearing as an imaginary part of the frequency) can be measured experimentally and then immediately compared to wave or turbulence models.

One may extend the frequencies from the real numbers (as oscillatory part) to
the complex numbers by including an imaginary part for a temporal damping. We
combine the excitation frequency

The hypothesis under consideration is that turbulent fluctuations are decaying
while being excited intermittently or continuously at different wavenumbers
to sustain the energy cascade balance. It is assumed that the fluctuations
are excited and subject to decay with a rate equivalent to the frequency
broadening

The Fourier transform of the wave field (which turns out to be the same as
the Laplace transform with respect to the decay rate

Breit–Wigner distribution for a half-value width of

The shape of the Breit–Wigner spectrum depends on two parameters: the
spectral peak frequency

The Breit–Wigner spectrum is tested against turbulent fluctuations in
interplanetary space. Four Cluster spacecraft

Sketch of the observational setup.

Time series data in the solar wind.

The streamwise wavenumber–frequency spectrum for the magnetic field
fluctuations is displayed in Fig.

Eulerian (spacecraft-frame) energy spectrum for magnetic field
fluctuations in the solar wind in the streamwise wavenumber–frequency domain.
The measurements of Cluster spacecraft on a time interval of 17 March 2005,
10:00–12:00 UT, are used. The spectra are displayed in physical units
(radians per kilometer for the wavenumber and radians per second for the frequency) in
the upper panel and in normalized units using the ion inertial length

The streamwise wavenumber–frequency spectrum is constructed as follows.
First, we collect the magnetic field data from each spacecraft on the
analyzed time interval and Fourier-transform the magnetic field fluctuations
(after subtracting the constant, mean field part) from the time domain into
the spacecraft-frame frequencies. The four frequency-domain magnetic field
data are put together into a 12

The wavenumber–frequency spectrum is obtained for the magnetic field data of
Cluster

For the least-squares fitting procedure, the Levenberg–Marquardt algorithm is
applied to each frequency slice on the logarithmic scale of the spectral
energy. The one-dimensional slices of the spectra and a fluctuation level of
the spectra are used as inputs to the least-squares fitting. A fluctuation
amplitude of 0.2 nT

One-dimensional slice of the energy spectrum over the
spacecraft-frame frequencies at a streamwise wavenumber of
0.015 rad km

The increasing sense of the peak frequencies and the half-value widths at
larger wavenumbers are quantitatively tracked by repeating the fitting
procedure at various wavenumbers (Fig.

The measured half-value widths

Peak frequency

The wave frequencies are corrected for the Doppler shift using the relation

Estimated peak frequencies in the plasma rest frame (corrected for the Doppler shift) as a function of the streamwise wavenumbers (data points in black), upper limit of the frequency estimate error (dashed line in gray), and dispersion relations for the whistler mode (middle curve) and the kinetic Alfvén mode (lower curve).

Decay rate is not constant over the spatial scales, but becomes increasingly larger (faster decay) toward smaller spatial scales. The increasing sense of decay at larger wavenumbers agrees with the picture of fluid turbulence, but the decay rate becomes increasingly larger than anticipated from the fluid turbulence models (e.g., random sweeping model, eddy turnover model). The larger values of the decay rate should be interpreted as a result of the nature of collisionless plasma turbulence.

Various scenarios are possible to explain the deviation of the decay rate
from the estimates based on fluid turbulence (random sweeping time or eddy
turnover time). First, wave–wave interactions are in operation and generate
other fluctuation types such as electric field or density fluctuations,
transferring the magnetic field fluctuation energy into the other energy
types. Second, wave-particle interactions are in operation, too, such as
coherent scattering (Landau or cyclotron resonance) or incoherent scattering
(pitch angle scattering). Indeed, the agreement with the whistler-mode
dispersion relation is indicative of the idea that the decay rate be
associated with the Landau or cyclotron damping rate of the linear mode.
However, it is interesting that the sudden increase of the decay rate is not
directly associated with the ion kinetic scales, since the wavenumber for the
ion inertial length (assuming protons) is about 0.016 rad km

To conclude the paper, we raise several issues that should be studied
more elaborately to strengthen (or disprove) the use of the Breit–Wigner
spectrum in plasma turbulence research.

The first is the issue of invariance of the spectral index of one-dimensional energy spectra between
the wavenumber domain

The spectral peak may break up into multiple branches when different linear mode waves are excited in the turbulent field at once. The frequency slice of the spectrum will show a convolution of different branches (with different peak frequencies and different half-value widths).

Another way to express the non-Gaussian shape distribution is to introduce higher-order moments. The fourth-order moment (in a non-trivial manner by measuring the deviation from a Gaussian shape, e.g., kurtosis or flatness index) is particularly suited to such a task. Perhaps there is a relation between the decay rate and the fourth-order moment of the frequency spread around the spectral peak.

The Cluster data are available at Cluster Science Archive,

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.

The work conducted in Graz, Austria, is financially supported by the Austrian Space Applications Programme at Austrian Research Promotion Agency (FFG ASAP-12) SOPHIE, the Solar Orbiter wave observation program in the heliosphere under contract 853994, and the Austrian Science Funds (FWF) Twisted magnetic flux ropes in the solar wind under contract P28764-N27. Yasuhito Narita acknowledges discussions with Uwe Motschmann and Karl-Heinz Glassmeier on the turbulence spectra. The work conducted in Braunschweig is financially supported by German Science Foundation under contract MO 539/20-1, DECODE: Detection of wave coupling cascade in space plasmas.Edited by: Bruce Tsurutani Reviewed by: Juan Valdivia and Peter Yoon