A review of analysis methods is given on quasi-monochromatic waves, turbulent fluctuations, and wave–wave and wave–particle interactions for single-spacecraft data in situ in near-Earth space and interplanetary space, in particular using magnetic field and electric field data. Energy spectra for different components of the fluctuating fields, minimum variance analysis, propagation and polarization properties of electromagnetic waves, wave distribution function, helicity quantities, higher-order statistics, and detection methods for wave–particle interactions are explained.
Waves and turbulence phenomena are observed in various regions of near-Earth space and interplanetary space such as in the solar wind, the foreshock, and the magnetosheath. Turbulence plays an important role in solar plasma (coronal heating and dynamo mechanism), collisionless shocks (particle acceleration), and interstellar space (diffusion of galactic cosmic ray). Due to its electrically conducting nature and collisionless nature, the picture of energy cascade of plasma turbulence is more diverse than that of fluid turbulence. Plasma physical processes such as wave–wave and wave–particle interactions serve as a channel of the energy cascade in addition to eddy splitting intrinsic to the fluid-like behavior of plasma. On kinetic scales on the order of the ion gyro radius (about 400 km in the solar wind at 1 astronomical unit from the Sun) or the electron gyro radius (about 10 km), waves become dispersive while interacting with individual particles (particle acceleration and scattering).
This paper is a review of analysis methods for waves, turbulence, wave–wave
interactions, and wave–particle interactions using in situ measurement data
of magnetic and electric fields. A summary of the analysis methods is
displayed in Table
Wave and turbulence analysis methods.
Turbulent fields may be composed of various fluctuation types such as linear mode waves, nonlinear wave components, and coherent structures. An overview of these fluctuation types is given here.
While magnetohydrodynamics (MHD) hosts three distinct linear wave modes
(fast, Alfvén, and slow modes), the kinetic treatment of plasma exhibits a
larger number of linear mode waves. Some are natural extensions of the MHD
modes, and the others are of a purely kinetic origin, resulting from the wave
resonance with individual electrons or ions. Kinetic wave modes from ion to
electron scales relevant to plasma turbulence (for oblique propagations to
the mean magnetic field) include the whistler mode, the ion Bernstein mode,
the kinetic Alfvén mode, the kinetic slow mode, the lower hybrid mode, the
electron cyclotron mode, the electron Bernstein mode, and the upper hybrid
mode (for a Maxwellian plasma). The dispersion relations are schematically
shown in Fig. Whistler mode is an extension of the MHD fast mode to the kinetic scales
(from the ion gyro scale down to the electron gyro scale) Ion Bernstein mode is of a strongly electrostatic nature, and appears as a
series of resonance break-ups of the whistler mode at the harmonics of the
ion cyclotron frequency in the limit to perpendicular propagation. Since the
ion Bernstein mode has higher frequencies than the ion cyclotron frequency,
the ion Bernstein mode can serve as “stations” of wave–wave couplings and
can sustain the daughter waves for a longer time, enabling a cascade of the
fluctuation energy to higher frequencies Kinetic Alfvén mode is a small-scale extension of the MHD Alfvén mode to
the ion-kinetic domain, and is obtained as a limit to perpendicular
propagation of the ion cyclotron mode. The sense of dispersion relation shows
a transition at a propagation angle of 70 to 75 Kinetic slow mode is a counterpart to the kinetic Alfvén mode, and is the
ion-kinetic extension of the MHD slow mode. The kinetic slow mode is of a
highly compressible nature, and is obtained as a quasi-perpendicular limit of
the low-frequency ion acoustic waves. The kinetic slow mode is only
moderately damped in the quasi-perpendicular directions (at angles around
85 Lower hybrid mode is of a strongly electrostatic nature and is a resonance
mode of the gyro motion of both the electrons and the ions. The resonance
frequency is about 43 times higher than the proton cyclotron frequency (it is
at Cyclotron modes (for ions and electrons) propagate in the parallel to oblique
directions to the mean magnetic field. The frequency rises up to the
cyclotron frequency of ions or electrons at which the wave electric field is
in resonance with the electron gyration. The resonance frequency becomes
lower at larger propagation angles. In the limit to perpendicular
propagation, the electron cyclotron resonance frequency falls down onto the
lower hybrid frequency. Electron Bernstein mode is of a strongly electrostatic nature, and appears at
frequencies close to the harmonics of the electron cyclotron frequency. Upper hybrid mode is a resonance mode as a result of the coupling of electron
gyro motion with the electron plasma oscillation. The frequency is higher
than the electron cyclotron frequency.
Schematic dispersion relations on ion-kinetic to electron-kinetic scales for oblique wave propagation to the mean magnetic field. Ion and electron cyclotron branches are associated with the propagate direction parallel or oblique to the mean magnetic field. The other branches are for oblique to quasi-perpendicular directions to the mean field. Whistler branches can exist for the quasi-parallel, oblique, and quasi-perpendicular directions.
Nonlinear modes can be any propagating wave components other than the linear mode fluctuations. Nonlinear modes may appear as large-amplitude solitary waves or as small-amplitude sideband waves at frequencies around that of the linear mode. The lifetime can be different and presumably depend on the fluctuation amplitude. Solitary waves may be stable if the wave steepening effect is balanced against the dispersion effect. Sideband waves may break into other frequencies and wavevectors through a successive wave–wave interaction.
Coherent structures appear in various forms, such as eddies, current sheets,
flux tubes, density cavities, or shocklets. Flux tubes may be twisted around
their axis, which can be deceptive to a circularly rotating wave. Coherent
structures are different from waves in that the coherent structures do not
propagate intrinsically, and appear as a zero-frequency mode in terms of wave
analysis. Formation of a thin current sheet leads to a hypothesis of electron
gyration-scale magnetic reconnection as an effective diffusion mechanism of
turbulent fluctuations
The role of coherent structures such as current sheets and possible
associated mechanisms such as magnetic reconnection should be further
highlighted. Coherent structures (particularly arising out of the turbulent
field) are considered ubiquitous in the free solar wind as well as in
magnetospheric plasma. Coherent structures populate signals in the solar wind
at a very high cadence, on scales on the order of the electron inertial
length, playing a role in the low-frequency fluctuations,
Quasi-monochromatic waves are identified as local peaks in the energy spectrum. For vectorial quantities such as the magnetic field, the electric field, and the flow velocity, the method of the spectral density matrix is particularly useful to extract the information on the wave properties.
The measured field data are Fourier transformed from the time domain into the
frequency domain:
The spectral density (SD) matrix
A detailed algorithm to evaluate the spectral power is given in
Notations for different coordinate systems useful in wave and
turbulence analysis: MFA (mean-field aligned),
Chopping the data must be treated carefully here because the chopping may
lead to violation of ergodicity. The ensemble averages, as in
Eq. (
The off-diagonal elements represent covariances of different fluctuation
components. In general, one may construct the covariance between one of the
field components (e.g., parallel magnetic field fluctuation to the mean
field) and the other fluctuation field (e.g., plasma density fluctuation). To
simplify the argument, the time factor
The SD matrix is conveniently analyzed by choosing the mean magnetic field as
the primary reference direction (mean-field-aligned system, MFA) to determine the wave
energy for perpendicular and parallel fluctuations (to the mean magnetic
field) and the field rotation sense around the mean field
The rotation sense of the field fluctuation is evaluated from the
off-diagonal elements of the SD matrix using the algorithm for the
ellipticity shown in Eqs. (
Ellipticity is evaluated through the angle
Magnetic field energy spectra for parallel and perpendicular fluctuating components to the mean magnetic field.
The SD matrix can be transformed into a diagonal form by using a unitary
matrix, and the wave properties are analyzed in the minimum variance system:
Different approaches are possible to determine the mean magnetic field from
the data, e.g., the mean values on the sub-intervals, the smoothing method,
the low-pass filtering method, and so on. How do we find the mean magnetic
field more properly if the mean field is no longer trivial, e.g., in a
scale-dependent or in a time-dependent fashion? The wavelet-based technique
Polarization ellipsoid in the minimum variance analysis.
The essence of the minimum variance analysis lies in the fact that the
minimum variance direction
For quasi-monochromatic electromagnetic waves, one may estimate the phase
speeds and the wavevectors from the electric and magnetic field data. The
phase speed is obtained as a ratio of the electric field amplitude to that of
the magnetic field:
From the phase speed
Wave distribution function is the concept of the wave energy distribution in
the wavevector domain assuming the existence of dispersion relations
(Fig.
Wavenumber–frequency diagram derived from the phase speed estimate.
The SD matrix is constructed from the electric field measurements in the
frequency domain. The SD matrix can on the other hand be modeled as a
projection of the wave polarization matrix (
Multi-spacecraft methods can be applied to multi-probe data analysis such as
the timing analysis to measure the phase speed or the wave telescope
technique or k-filtering technique to measure the fluctuation energy in the
wavevector-frequency domain
Wave distribution function
Frequencies in the observer's (or spacecraft) frame are a sum of the
intrinsic wave frequency, modulation of the intrinsic frequency due to the
random sweeping effect by the large-scale flow velocity fluctuation or the
nonlinear (sideband) effect, and the Doppler shift by the mean flow:
Using the Stokes parameters
Stokes spectrum decomposing the fluctuations into circularly rotating fields.
The Elsässer variables are additive couplings of the magnetic field with
the flow velocity by adapting the dimension of the magnetic field into that
of the velocity:
The Elsässer variables give an intuitive picture of magnetohydrodynamics
that parallel propagating Alfvén waves (to the mean magnetic field) are
expressed by
Helicity quantities play an important role in turbulence
The magnetic helicity density is defined using the vector potential
Elsässer variable spectrum.
Structure of the spectral density matrices for magnetic field and flow velocity data.
The cross helicity density represents a covariance between the flow velocity
and the magnetic field,
Wave–wave couplings can be measured by extending the notion of covariance to
multiple wave components. A three-wave coupling, for example, occurs under
the condition of frequency and wavenumber conservations:
Three-wave coupling diagram for Alfvén wave scattering (left) and
parametric decay of Alfvén waves into Alfvén and sound waves (right).
Figure adapted from
Charged particles can exchange the energy with the wave electric field both
parallel to the mean magnetic field (Landau resonance) and perpendicular to
the mean field (Landau resonance). Figure
Wave–particle interactions and the associated part of the velocity distribution functions.
Charged particles can be scattered by the wave electric and magnetic fields
incoherently, and the scattering deforms the velocity distribution function
along the co-centric contours centered at the wave phase speed
(Fig.
Revealing the fluctuation properties is essential to advance our knowledge on turbulent plasmas using spacecraft data. In the following, particularly challenging questions are addressed that should be focused on for the upcoming spacecraft missions.
“What is the role of dispersion relation in turbulence?” Whether a dispersion relation exists in a turbulent field is a very important
problem to guide to a theory of turbulence. One of the pictures of turbulence
development is a transition from linear mode waves into more randomized
nonlinear waves through the breakdown of the dispersion relation. The
analysis of dispersion relation diagram is possible both from
single-spacecraft and multi-spacecraft data. Perhaps the appearance of linear
mode waves depends on the evolution time from the instability onset or the
fluctuation amplitudes. “What are the intrinsic spectra of turbulence?” Turbulence is essentially a spatially and temporally developing phenomenon,
and the energy spectra must be viewed as a 4-D quantity as a function of the
frequencies and the three components of the wavevectors. Turbulent
fluctuations appear in the magnetic field, the electric field, and the plasma
fluctuations such as the flow velocity, the density, and the temperature.
Moreover, the magnetic and electric fields are vectorial quantities and a
more complete picture of the spectra needs to be constructed using the SD
matrices. As a consequence, a large number of spectra are necessary to
characterize the turbulent fields unambiguously. “How random are the wave phases in turbulence?” Turbulent fields cannot have fully random phases, since otherwise the
constituent waves (or fluctuations) cannot interact with one another and the
energy cascade through wave–wave interactions becomes impossible. The
elementary energy transport process can occur only as a coherent process
under the resonance conditions for the frequencies and the wavevectors. On
the other hand, turbulent fluctuations are apparently incoherent. Otherwise
the superposition of individual waves ends up with a large-scale coherent
structure. The cascading waves generated by the wave–wave coupling attain a
more random phase at some stage of evolution.
Wave analysis methods assume that the measured fluctuation data are cleaned against noise or spacecraft-generated disturbance by proper calibration procedures. The signals in the data must clearly be identified to separate them from the noise. In the case of magnetometer data, the offsets and the noise floor must be determined prior to the data analysis.
Turbulent fields may contain coherent structures with eddies, current sheets, and discontinuities. Coherent structures can be conveniently studied by, for example, introducing the de Hoffmann–Teller frame for the shock waves (eliminating the convective electric field with the sliding frame along with the plane of the discontinuity), the analysis of electrostatic potential through the Liouville mapping, or visualization of the magnetic field and the plasma distribution using the Grad–Shafranov equation for a magnetohydrodynamic quasi-equilibrium state.
The fluctuations can also be highly intermittent such that the small-scale burst-like fluctuations are more localized and the fluctuation statistics strongly deviate from a Gaussian process. Various methods have been developed to characterize the intermittency, such as the probability density function, the local intermittency measure, the multi-fractal method, the partition function, and the partial variance increments, wave phase shuffling, and surrogation.
No data sets were used in this article.
The authors declare that they have no conflict of interest.
This work is financially supported by the Austrian Space Applications Programme at the Austrian Research Promotion Agency, FFG ASAP-12 SOPHIE (Solar Orbiter wave observation program in the heliosphere) under contract 853994. The author acknowledges discussions with the Science Study Team for the ESA THOR mission concept. Edited by: V. Carbone Reviewed by: two anonymous referees