Space–time structure and wavevector anisotropy in space plasma turbulence
Abstract
Space and astrophysical plasmas often develop into a turbulent state and exhibit nearly random and stochastic motions. While earlier studies emphasize more on understanding the energy spectrum of turbulence in the onedimensional context (either in the frequency or the wavenumber domain), recent achievements in plasma turbulence studies provide an increasing amount of evidence that plasma turbulence is essentially a spatially and temporally evolving phenomenon. This review presents various models for the space–time structure and anisotropy of the turbulent fields in space plasmas, or equivalently the energy spectra in the wavenumber–frequency domain for the space–time structures and that in the wavevector domain for the anisotropies. The turbulence energy spectra are evaluated in different onedimensional spectral domains; one speaks of the frequency spectra in the spacecraft observations and the wavenumber spectra in the numerical simulation studies. The notion of the wavenumber–frequency spectrum offers a more comprehensive picture of the turbulent fields, and good models can explain the onedimensional spectra in the both domains at the same time. To achieve this goal, the Doppler shift, the Doppler broadening, linearmode dispersion relations, and sideband waves are reviewed. The energy spectra are then extended to the wavevector domain spanning the directions parallel and perpendicular to the largescale magnetic field. By doing so, the change in the spectral index at different projections onto the onedimensional spectral domain can be explained in a simpler way.
Keywords
Dispersion relation Anisotropy Solar wind turbulence1 Introduction
Plasmas in space and astrophysical systems often develop into a turbulent state. Examples of turbulent plasmas and magnetic fields can be found in the solar system, too, such as solar atmosphere (photospheric convections, formation of complex magnetic network), interplanetary space (solar wind flow and interplanetary magnetic field), and planetary magnetospheres (shockupstream and shockdownstream regions, magnetotail region). Magnetic field generation or amplification is possible in the solar and planetary interior by turbulent dynamo processes such as the twisting effect on the magnetic field in a turbulent flow (often referred to as the alpha effect for the Sun, and in general, convective effects are at work in planetary interiors). Understanding plasma turbulence has also immediate implications on the problems of coronal heating, acceleration and transport of galactic cosmic ray, and onset of magnetic reconnection process.
Turbulent fluctuations represent nearly random pattern or motion of the flows, and can be found in our daily experience to geophysical scales such as turbulence in the ocean or in the atmosphere. Our modern understanding of turbulence owes a lot to the picture of energy cascade developed by Richardson (1926) and later formulated as a powerlaw spectrum for a realization of the inertial range by Kolmogorov (1991). Application of the renormalization method to fluid dynamics was also a success. The LagrangianHistory Direct Interaction Approximation (LHDIA) developed by Kraichnan (1965b) is a demonstration that one can derive Kolmogorov’s inertialrange spectrum from Navier–Stokes equation without introducing any adjustable parameter.
There are two pillars in physics of fluid turbulence. One is the scale invariance and the other is isotropy. Both of them are the properties of Navier–Stokes equation in the inviscid limit (zero viscosity limit), e.g., Frisch (1995). These two properties are related to the scaling symmetry and the rotation symmetry in the fluid system. Turbulence occurs whenever a fluid motion satisfies certain conditions, and occurs independently from types of fluid (gas or liquid). Excitation of randomlyoriented and randomlysized eddies can be recognized in artistic paintings or drawings. A review by Warhaft (2002) gives an intuitive comparison between randomsize and similarsize phenomena. Formation of eddies can be confirmed in many experimental setups for turbulence measurements, e.g., turbulent boundary layer experiment (Falco 1977).
There are two unique properties in the astrophysical plasmas that make studies of turbulence particularly challenging. The first effect is the coupling with electromagnetic fields. Plasmas represent an ionized gas, and are electrically conducting. Gas dynamics and electromagnetism must be coupled to each other in dynamics. While the inertial range of fluid turbulence essentially represents splitting of eddies toward smaller spatial scales, that of plasma turbulence may represent the energy transport mediated by electromagnetic waves (in addition to eddies). The second effect is the collisionless nature of astrophysical plasmas. The number density is so low (or plasmas are so dilute) that the particle mean free path reaches the same order of the system size. Binary collisions of particles are rare, and energy dissipation in plasma turbulence cannot be explained merely by a diffusion process. Instead, the energy dissipation needs to be explained by wave–particle interactions, i.e., by exchanging energy mediated by the wave electric field.
It is worth mentioning that the onedimensional energy spectra are typically derived in the spacecraftframe frequency domain in the observations, while the spectra from numerical simulations (using a magnetohydrodynamic code, a hybrid plasma code, or a particleincell code) are often estimated in the wavenumber domain as illustrated in Fig. 1. Spacecraft data are obtained as time series data, and the Fourier transform of the time series data are obtained in the frequency domain. Simulation data are often stored at snapshots over the spatial coordinates at discrete time steps. The Fourier transform of the simulation data in the spatial coordinates are obtained in the wavenumber or wavevector domain. Both the spectrum in the frequency domain and that in the wavenumber domain exhibit a power law. That is, the frequency spectrum shows a power law \(E(\omega ) \propto \omega ^{\alpha _{\omega }}\) and the wavenumber spectrum also shows a power law \(E(k) \propto k^{\alpha _{k}}\). Moreover, the spectral indices \(\alpha _{\omega }\) and \(\alpha _{k}\) are often the same or sufficiently close to each other. One may then introduce Taylor’s frozenin flow assumption (Taylor 1938) and interpret the frequencies as the streamwise wavenumbers, for example, a spectral index of \(\,5/3\). How can we explain this fact? In the presence of finite wave propagations, however, the mapping between the frequency spectra and the wavenumber spectra is no longer unique. The frequency spectra and the wavenumber spectra need to be understood as different projections of the energy spectrum in a higher dimensional sense spanning both the wavenumbers and the frequencies.
2 Space–time structure
2.1 Hydrodynamic picture
 1.The Gaussian frequency distribution \(F(\omega , \mathbf {k})\) reduces to the Dirac delta function in the limit of vanishing sweeping velocity, \(\delta U \rightarrow 0\):$$\begin{aligned} F(\mathbf {k}, \omega )= & {} \frac{1}{\sqrt{2\pi k^2 (\delta U)^2}} \exp \left[ \frac{(\omega  \mathbf {k} \cdot \mathbf {U}_0 )^2}{2 k^2 (\delta U)^2} \right] \end{aligned}$$(11)such that the streamwise wavenumber–frequency spectrum (in the direction of the mean flow) reduces to$$\begin{aligned}\rightarrow & {} \delta (\omega  \mathbf {k}\cdot \mathbf {U}_0) (\delta U \rightarrow 0) \end{aligned}$$(12)Equation (13) is nothing other than Taylor’s frozenin flow hypothesis (Taylor 1938) in the spectral domain, a onetoone projection or mapping of the frequencies onto the streamwise wavenumbers using the mean flow speed \(U_0\).$$\begin{aligned} E(k_\mathrm{flow}, \omega ) = E(k_\mathrm{flow}) \delta (\omega  k_\mathrm{flow}\cdot U_0) . \end{aligned}$$(13)
 2.For an infinitely long inertialrange spectrum in the onedimensional wavenumber domain, \(E(k) \propto k^{\alpha }\), the spectral index \(\alpha \) of the onedimensional spectrum is invariant between the frequency domain and the wavenumber domain regardless the choice of the mean flow speed \(U_0\) and the largescale flow speed variation \(\delta U\). The frequency spectrum exhibits the same powerlaw as that in the wavenumber domain with a difference from the wavenumber spectrum only in the coefficient. The energy spectra in the wavenumber domain and the frequency domain are$$\begin{aligned} E(k)= & {} C_\mathrm{K} \epsilon ^{2/3} k^{5/3} \end{aligned}$$(14)where the coefficient on the frequency spectrum C(U, V) is given by$$\begin{aligned} E(\omega )= & {} C(U_0,\delta U) C_\mathrm{K} \epsilon ^{2/3} \omega ^{5/3} \,, \end{aligned}$$(15)The symbol \(C_\mathrm{K}\) denotes the Kolmogorov constant, and \(\epsilon \) the energy dissipation rate.$$\begin{aligned} C(U_0,\delta U) = \int _0^\infty \mathrm{d}\gamma \frac{\gamma ^{2/3}}{4U_0} \left[ \mathrm{erf}\left( \frac{\gamma + U_0}{\sqrt{2}\, \delta U} \right)  \mathrm{erf}\left( \frac{\gamma  U_0}{\sqrt{2}\, \delta U} \right) \right] . \end{aligned}$$(16)
 3.The Lagrangianframe frequency spectrum can also be obtained when using the Richardson–Kolmogorov scaling \(\delta U \sim (\ell \epsilon )^{1/3}\):$$\begin{aligned} E(\omega ) = C_\mathrm{L} \epsilon \omega ^{2} . \end{aligned}$$(17)
2.2 Magnetohydrodynamic picture
2.2.1 Is turbulence strong or weak?
Turbulence serves as a source of energy for the smallscale fluctuations, while serving as a sink of energy for the largescale structures, e.g., the Reynolds stress tensor in the meanfield dynamics. One speaks of turbulence being “strong” such that the fluctuating field influence the mean field and the largescale structure becomes deformed or even destroyed by the fluctuating field, and turbulence being “weak” such that the turbulent fluctuations are mostly composed of linearmode waves.
In the strong turbulence treatment, nonlinearities are so effective that fluctuation amplitudes alter the mean magnetic field, making the mean field inhomogeneous. Both nonlinearities and inhomogeneities need to be taken into account to describe strong turbulence. Various models and concepts have been developed in order to treat strong turbulence theoretically, e.g., mixing length, eddy viscosity, Alfvén time, k\(\epsilon \) model (Biskamp 2003). Applications of the strong turbulence approach to solar wind turbulence are found in Yokoi (2006); Yokoi and Hamba (2007); Yokoi et al. (2008).
2.2.2 Wave approach
The Doppler shift has a form of linear dispersion relation. If the turbulent medium has not only a mean flow but also largescale linear mode waves such as Alfvén waves in a magnetofluid, the energy spectrum exhibits a Dopplershifted dispersion relation. Furthermore, if there are multiple wave modes or dispersion relations, the spectrum splits into the respective dispersion relations.
2.3 Kinetic waves
Dispersion means a broadening of wave packet in the direction of propagation. Dispersion is caused by the frequencydependent (or wavenumberdependent) group speed. Thus, waves are dispersive when the dispersion relation is curved in the wavenumber–frequency domain. An example is the parallelpropagating whistler mode (to the mean magnetic field). At very low frequencies (far below the ion gyrofrequency) the whistler mode behaves as an MHD fast mode and the dispersion relation is linear between the frequencies and the wavenumbers, \(\omega \propto k\). The group speed \(v_\mathrm {gr} = \partial \omega / \partial k\) is independent from the wavenumbers. At intermediate frequencies around the ion gyrofrequency and higher, the whistler frequencies depend on the wavenumbers quadratically, \(\omega \propto k^2\). The group speed depends now on the wavenumbers, and becomes higher at larger wavenumbers.
The curved shape of the dispersion relation indicates that the ratio of the electric field to the magnetic field also depends on the wavenumbers, since the ratio is closely related to the phase speed, \(v_\mathrm {ph} = \omega /k = \delta E/ \delta B\), due to the induction equation. The electric field amplitude becomes increasingly larger for a dispersive wave with an increasing sense of frequencies like the whistler mode. In a dispersive plasma, the fluctuation energy can be transferred from the magnetic field onto the electric field, or vice versa.
Dissipation means a temporal damping of fluctuating fields by binary collisions, finite resistivity, or wave–particles interactions. In a collisionless plasma, wave–particle interactions can occur in the longitudinal sense (Landau resonance) by accelerating particles that have a similar velocity with the wave phase speed (such that the interaction time becomes longer) as well as in the transverse sense by accelerating particles through a resonance in the cyclotron motion. Dissipation is essentially a heating process, and is considered as irreversible. Whistler waves at frequencies close the electron gyrofrequency can be in resonance with the electrons and are subject to the cyclotron damping.
The ion gyroradius and the inertial length are of the order of several hundred to 1000 km under a typical condition of the solar wind at the Earth orbit (1 astronomical unit from the Sun), while that of the electrons are of the order of ten to hundred km. The picture of linear modes is useful in understanding fluctuations in space plasma. The ionkinetic waves can be regarded as extensions of the MHD linear wave modes to smaller wavelengths, and can be grouped into the Alfvén mode, the fast mode, and the slow mode families.
Each kinetic mode is obtained by a linearly perturbing the Vlasov equation (typically assuming a Maxwellian or a biMaxwellian plasma) and solving the equation for the wave electric field (which reduces to finding nontrivial roots for the wave dielectric tensor) under a given set of parameters like the wavenumber, the propagation angle to the mean magnetic field, the plasma parameter beta, the iontoelectron temperature ratio, and the Alfvén speed with respect to the speed of light (Stix 1992; Gary 1993). The solution is obtained in the form of dispersion relation, that is, the frequency dependence in the complex number domain as a function of the wavenumbers or wavevectors. Analytic solutions are obtained only in few cases.
For a quasiparallel propagation to the mean magnetic field, possible ionkinetic modes (assuming a temperature isotropy) are the whistler mode, the ion–cyclotron mode, and the ionacoustic mode. For a quasiperpendicular propagation, there are four possible ionkinetic modes as a transition of the quasiparallel modes: the kinetic Alfvén mode, the kinetic slow mode, the oblique whistler mode, and the ion Bernstein mode. Those wave modes can be grouped into the Alfvén, fast, and slow mode families by tracking the dispersion relation onto the largescale magnetohydrodynamic modes as follows.
2.3.1 Alfvén mode family
Ion cyclotron mode
Kinetic Alfvén mode
2.3.2 Fast mode family
Whistler mode
Ion Bernstein mode
Lower hybrid mode
2.3.3 Slow mode family
Ion acoustic mode
2.4 Zerofrequency mode
The zerofrequency mode represents a nonpropagating perturbation of density and temperature (Kadomtsev 1960). The entropy mode does not change the total plasma pressure nor the specific entropy on the perturbation Since the frequencies are zero at various wavenumbers, the propagation speeds (phase speed) are also zero in the plasma rest frame (comoving with the mean flow).
2.5 Sideband waves
The energy spectrum in the wavenumber–frequency domain shows the sideband wave activity whenever the condition for the ideally convection (Doppler shift) or that for the plane waves breaks down. The sideband waves have a frequency deviation from the linear mode dispersion relation, and the energy spectrum shows a spread around the spectral peak. The Doppler broadening is one example, but there are a variety of reasons and mechanisms for the excitation of the sideband waves: the random sweeping by the largescale flow variations or the largescale waves (both cause the Doppler broadening), the excitation of sideband waves by wave–wave interactions, the wave damping effect, and the wave packet formation.
2.6 Coherent structures
Turbulent fluctuations are not fully random but must necessarily have both random phases and coherent phases. For example, the resonance conditions for the wave–wave couplings [Eqs. (45)–(46)] a coherent process because the phase of the generated wave is automatically determined by that of the initial waves, including the initial phase. Coherence in the phase results in a structure formation. Coherent structures appear as nonpropagating, standing structures in the turbulent fields, and are merely swept by the mean flow.
There are a variety of coherent structure types. Eddies are the fundamental constituent of hydrodynamic turbulence and originate in the advection term of the Navier–Stokes equation. If the medium is compressible, shock waves or density cavity can occur. In the case of plasma turbulence, onedimensional current sheet can be formed with various thicknesses down to the electron gyroradius. If the current sheet is sufficiently thin, magnetic reconnection sets on and generates bursty flows. The electric current and the magnetic field can confine the plasma and form flux tubes, flux ropes, and forcefree type spiral magnetic field structures. In the shockdownstream region such as the magnetosheath region of planetary magnetospheres, the mirror instability sets on due to the overheated plasma in the perpendicular direction to the mean magnetic field and the pressure balanced structure is formed, making a balance between the thermal pressure and the magnetic pressure.
2.7 Lessons from the observations
2.7.1 Eulerian picture
2.7.2 Catalogue of dispersion diagrams

Using multispacecraft data, the dispersion relation diagrams are obtained by determining frequencies from the time series data (with the help of Fourier transform) and wavevectors from the phase difference from an sensor to another. There are various ways to efficiently determine the wavevectors from multispacecraft data, e.g. , the timing or phase differencing method (Hoppe and Russell 1983; Dudok de Wit et al. 1995), the minimum variance projection such as the wave telescope or kfiltering projection (Capon 1969; Neubauer 1990; Pinçon and Lefeuvre 1991; Motschmann et al. 1996; Glassmeier et al. 2001), and the eigenvaluebased projection (Schmidt 1986). Also, the correction for the Doppler shift \(\mathbf {k} \cdot \mathbf {U}\) is possible once the wavevector \(\mathbf {k}\) and the bulk flow velocity \(\mathbf {U}\) are known to interpret the frequencies in the rest frame of plasma, comoving with the bulk flow.

Using the electric and magnetic field data, one computes the ratio of the electric field to the magnetic field. This ratio is a phase speed (in the observer frame) when Fourier transforming the induction equation, \(v_\mathrm {ph} = \omega /k = \delta E_\mathrm {tr1} / \delta B_\mathrm {tr2}\). See, e.g., Bale et al. (2005) or Eastwood et al. (2009) for applications. It is important to note here that two mutuallyorthogonal transverse components to the wave propagation direction (wavevector direction) need to be used to estimate the phase speed and that one assumes only one wave mode at a given frequency (i.e., only one dispersion branch). Otherwise the estimate of the phase speed may be mixed up with the electrostatic component or be influenced by multiple dispersion branches.
Figure 10 displays nine samples of dispersion relation diagram in the solar wind on spatial scales around ion kinetic motion (around ion inertial length, typically at about 400 km). The diagrams are obtained by the following procedure. Fourpoint magnetic field data from the Cluster spacecraft are used here. The magnetic field data are Fouriertransformed from the time domain into the spacecraftframe frequency domain for each spacecraft, and are further projected from the spatial domain onto the wavevector domain to obtain the \(3\times 3\) spectral density matrix as a function of spacecraft frequencies and wavevectors. The projection of the fluctuating fields onto the wavevector domain is achieved by a combination of the minimum variance projection with the eigenvaluebased decomposition the multipoint signal resonator technique (Narita et al. 2011). The local peaks of the energy spectra in the threedimensional wavevector domain are identified at each frequency (in the spacecraftframe of reference) by scanning the total fluctuation energy (trace of the spectral density matrix) and a set of the frequencies and the wavevectors are obtained. Fluctuating fields are assumed to be composed of a set of plane waves and noise without assuming or imposing any dispersion relation in the data. The frequencies are transformed from the spacecraft frame into the plasma rest frame comoving with the mean bulk flow speed by correcting for the Doppler shift. The restframe frequencies are normalized to the ion cyclotron frequency as \(\omega _\mathrm {re}/\varOmega _\mathrm {i}\), and the wavenumbers (magnitude of the wavevector) to the ion inertial wavenumber as \(k V_\mathrm {A}/\varOmega _\mathrm {i}\). Finally, the dispersion relations for theoretically predicted linear modes are overplotted using the values of propagation angle \(\theta _\mathrm {kB}\) to the mean magnetic field (averaged over the wavevector domain) and ion beta from the measurements. Propagation directions are highly oblique (almost perpendicular) to the mean magnetic field in the solar wind. The detected wave components (a set of the frequencies and the wavenumbers) are associated with different wave modes: kinetic Alfvén mode (in circles), ion Bernstein mode for protons (in triangles) and for helium alpha particles (in diamonts), and nonlinear or sideband mode (in squares). Dispersion relations including a variation or an uncertainty of propagation angles are also displayed for kinetic Alfvén mode (the lowest frequencies), heliumalpha Bernstein mode (the second lowest at a half of the proton cyclotron frequency), proton Bernstein mode at the proton cyclotron frequency and the second harmonic.
The detected waves are associated to different linear modes including uncertainties in the Doppler shift correction. The detected waves that have large deviations in frequency from the linear mode ones are grouped into the sideband waves or nonlinear waves. While about 25% of the wave population is associated with the kinetic Alfvén mode and another 25% with the ion Bernstein mode for protons, about 40% of the wave population are outside the frequency ranges expected from the linear Vlasov theory and represent the sideband waves. Dispersion analysis shows various examples of linear mode waves: kinetic Alfvén mode (Sahraoui et al. 2010; Roberts et al. 2015a), ion Bernstein modes (Perschke et al. 2013), whistler mode (Narita et al. 2011). Whether those “offbranch” waves are instantaneous waves generated by wave–wave interactions (which are presumably damped quickly) or a fragment of propagating coherent structure remain unsolved. A test for phase coherence or a study of waveform will be helpful to understand the physical process of the sideband or offbranch waves in the observations.
2.7.3 Statistical dispersion diagram
3 Wavevector anisotropy
3.1 Impact of the largescale magnetic field
Plasma turbulence is intrinsically anisotropic whenever the largescale magnetic field is present. On the individual particle level, the Lorentz force (\(q \mathbf {v}\times \mathbf {B}\), where q is the electric charge, \(\mathbf {v}\) the particle velocity, and \(\mathbf {B}\) the magnetic field) acts on the charged particle in the perpendicular components of the velocity to the magnetic field. On the fluid scale, the Lorentz force (\(\mathbf {j} \times \mathbf {B}\)) plays an important role in the plasma dynamics, contributing as the magnetic pressure gradient force and the magnetic tension. The anisotropic nature in the plasma dynamics also influences the energy transfer process in turbulence from one scale to another as well as the structure formation in the turbulent field.
Homogeneous and isotropic fluid turbulence is, in contrast, essentially composed of eddies with the vorticity axes in various directions and in various magnitudes. The turbulence energy transport is carried by the interactions between eddies generating eddies again with the vorticity axes in various directions and in various magnitudes. The interaction of eddies does not recognize the largescale structure (except for a situation near the boundary or the wall of turbulent flow) in the inertial range.
Plasma turbulence has a larger degree of freedom (or control parameters) in that not only eddies but also electromagnetic waves such as Alfvén waves can interact with one another, and the wave–wave interactions are an additional energy carrier for the turbulent cascade. Electromagnetic waves in the plasma are so diverse. The wave mode or the dispersion relation is a function of the plasma parameter beta and the propagation angle. Furthermore, the fluctuation sense is also diverse such as the righthanded or lefthanded rotation sense of the wave field and the compressible or incompressible sense of fluctuation with respect to the magnetic field direction.
Anisotropy may enter plasma turbulence in various ways, e.g, in the energy spectra, in the fluctuating sense, in the energy transfer rate, and in the dissipation rate. We limit ourselves this section to the anisotropic structure formation in turbulence. Anisotropy appears as extended or elongated structures of the energy spectrum in the wavevector domain spanning the parallel and the perpendicular components to the mean magnetic field (assuming, for simplicity, that the mean field can nearly be regarded as constant). Even with single spacecraft measurements, there are indications that the energy spectrum be anisotropic with respect to the mean magnetic field, for example, the change in the spectral index as a function of the projection angle (or the flow direction) to the mean magnetic field in Fig. 2 (Horbury et al. 2008; Osman and Horbury 2009) or the change in the correlation length with respect to the mean field (Matthaeus et al. 1990; Dasso et al. 2005).
3.2 Twocomponent model
The presence of the largescale or mean magnetic field causes two particular effects in plasma dynamics, which leads to a picture of two competing fluctuation geometries in the wavevector domain.
Perpendicular cascade scenario
The picture of the perpendicular wavevector geometry is constructed by considering threewave couplings for the Alfvén waves (Shebalin et al. 1983; Biskamp 2003). The threewave resonance conditions in the frequencies \(\omega \) and the wavevectors \(\mathbf {k}\) are expressed in Eqs. (45) and (46), respectively. If one imposes that all the participating waves are the Alfvén waves with the dispersion relation \(\omega = \mathbf {k} \cdot \mathbf {V}_\mathrm{A}\), the system of equations results in that the parallel wavevector component is zero for one interacting or participating wave, and is the same between the other participating wave and the generated wave such that \(k_{\Vert (a)} = 0\) or \(k_{\Vert (b)} = 0\). Since the frequency of the Alfvén wave is zero for the perpendicular propagation, one interacting wave component is a nonpropagating spatial structure. The generated wave (wave component c) has a larger perpendicular wavevector component, and the cascade can proceed in the perpendicular direction. Two counterpropagating Alfvén waves as in the magnetohydrodynamic turbulence phenomenology by Iroshnikov (1964) and Kraichnan (1965a) cannot interact with each other in a threewave coupling order, but at least in a fourwave coupling order or higher. If the cascade continues in the perpendicular direction, the wavelengths become smaller across the largescale magnetic field while the wavelengths do not change in the parallel direction. The perpendicular cascade causes the formation of filament structures in plasma.
Parallel cascade scenario
Energy cascade is possible in the parallel direction under various conditions, for example, when four waves are even more waves are involved in the wave–wave interactions or when the conversion is possible into other wave modes such as the sound mode or dispersive modes. In the fourwave interactions, the wavevectors of the generated waves cannot be determined uniquely (and the parallel component of the wavevector is no longer constant). If the mode conversion is possible, a largeamplitude Alfvén wave collapses in a threewave coupling sense into a forwardpropagating sound wave (with respect to the direction of the original Alfvén wave) and a backwardpropagating Alfvén wave, known as the decay or modulational instabilities (Derby 1978; Goldstein 1978). An Alfvén wave can also collapse into two forwardpropagating daughter waves, (Mio et al. 1976; Mjølhus 1976; Nariyuki and Hada 2006). The generated waves do not have to lie on the dispersion relation. If the dispersion relation is not linear (straight line) but dispersive (curved line) and the propagation speed is frequencydependent, the generated wave may happen to be on the dispersion relation. The decay and the modulational instabilities are systematically and numerically studied (Longtin and Sonnerup 1986; Terasawa et al. 1986; Wong and Goldstein 1986) in view of Hallmagnetohydrodynamics in which circular polarized largeamplitude Alfvén waves are obtained as an exact solution
One of the useful approximations is to regard turbulent fluctuations as a superposition of the fluctuation component for parallelpropagating waves (to the mean magnetic field, referred to as the slab geometry) and that for perpendicular wavevectors (referred to as the quasitwodimensional turbulence geometry). In solar wind turbulence, the fluctuation geometry for quasitwodimensional turbulence is estimated to have larger fluctuation amplitudes (Bieber et al. 1996). The dominance of quasitwodimensional turbulence is also indicated in the study of cosmic ray transport (a long meanfreepath of cosmic ray diffusion) (Bieber et al. 1994). On the other hand, the dominance of the slab or the quasitwodimensional fluctuation geometry can be casedependent such that the lowspeed stream in the solar wind is characterized by the quasitwodimensional turbulence geometry and the highspeed stream by the slab geometry (Dasso et al. 2005).
3.3 Critical balance model
The critical balance hypothesis successfully explains the change in the spectral slope as a function of the projection angle from the mean magnetic field (Forman et al. 2011). The scaling relation is found to be valid in numerical simulations (Cho and Vishnian 2000) and the relation between the wavevector components is also confirmed in the inner heliosphere (He et al. 2013).
3.4 Elliptic anisotropy model
Numerical simulation of MHD turbulence also supports the anisotropic energy spectrum. A nearly elliptic shape of the energy spectrum is obtained from magnetohydrodynamic turbulence simulation (Shebalin et al. 1983) in which the initial spectrum is set to isotropic, and anisotropy develops in such a way that the spectrum extends perpendicular to the largescale magnetic field forming an elliptic shape. Elliptic sense of the wavevector anisotropy is the simplest and a natural extension of the energy spectrum from the isotropic case to an anisotropic one. The reason for this lies in the fact that the elliptical shape appears in the lowestorder (secondorder) polynomial expansion of a smooth function in the twodimensional domain such as a space–time correlation function (He and Zhang 2006).
3.5 Nonelliptic anisotropy model
3.6 Asymmetries
The energy spectrum may appear asymmetrically in two different ways with respect to the direction of the mean magnetic field. One is the energy imbalance between the parallel and the antiparallel direction to the mean field, and the other is an asymmetry in the azimuthal directions around the mean field.
Axial asymmetry is indicated by in situ measurements in the solar wind. Using single spacecraft data and a mapping procedure, the threedimensional structure of solar wind turbulence is obtained from the Ulysses spacecraft during a polar pass at the heliocentric distance 1.4–2.6 AU in 1995 (Chen et al. 2012). The fluctuations are axially asymmetric in the directions around the mean magnetic field. Using multispacecraft data, the axially asymmetric energy spectrum is presented directly in the threedimensional wavevector domain (Narita 2014). At the time of the manuscript writing, there is no direct clue as to the origin or the mechanism of the axial asymmetry. The mechanism may stem from an asymmetric radial flow expansion in the heliosphere or an intrinsic plasma process.
3.7 Lessons from the observations
Observationally speaking, it is possible to estimate, using some assumptions, the energy spectrum in the wavevector domain by two different mapping methods from the frequency domain onto the wavevector domain.
4 Outlook

applications and limits of Taylor’s hypothesis

Doppler shift and Doppler broadening

existence of dispersion relations

energy cascade directions

visualization of of anisotropies and asymmetries
Control parameters
While the plasma parameter beta is often used as a control parameter in (roughly) determining the behavior of plasma dynamics, e.g., dispersion relations, there may be more control parameters. The fluctuation amplitude with respect to the mean field (for example, magnetic field) may be regarded as an index for the strength of nonlinearity. In fact, the notion of linear modes is valid only when the fluctuations are sufficiently small such that the wave–wave interactions are negligible. In reality, many waves including linear mode waves and nonlinear or sideband waves can be excited in plasma turbulence.
Transition into fullydeveloped turbulence
The frequency broadening around the Doppler shift (which appears as the Doppler broadening) and the dispersion relations (which appears as the sideband waves) has different possible origins and may plan an important role in turbulence evolution. While the fluctuation of the largescale flow (or the random sweeping effect) causes the Doppler broadening, wave–wave coupling causes the sideband waves around the normal mode dispersion relations. Whether the waves generated by the threewave coupling are supported by the normal mode can be studied by the mismatch test as in Gary (2013). One may, for example, regard the sideband waves as a proxy of turbulence evolution degree (how much the fluctuations are stored on the normal modes and the sideband waves) as in Comişel et al. (2013). The threewave coupling model is illustrative and in that the model can combine different components such as the linear mode waves, the sideband waves, and the wavevector anisotropies. The lifetime of the sideband waves should be measured or studied in more detail to reveal the transition into turbulence.
Mechanism causing axial asymmetry
The axial asymmetry remains one of the unsolved problems in plasma turbulence. Both single spacecraft and multispacecraft measurements show evidence for the axial asymmetry. Naively speaking, there are two causes, one by intrinsic processes of plasma turbulence and the other by external processes such as an inhomogeneous flow or an spatially expanding flow. In the linear Vlasov theory treatment, axial symmetry is explicitly broken by selecting the direction of wave propagation (or the wavevector direction) and the coordinate system is constructed using the direction of the largescale magnetic field and the direction of the wave propagation. Axial symmetry is already broken when deriving the normal mode dispersion relation. In the former scenario, since the fundamental equations are invariant under the rotation around the magnetic field direction, it is natural to anticipate the Nambu–Goldstone mode that compensates for the broken symmetry by exciting an oscillation in the sense of the original symmetry. This argument leads us to predict that the wavevector anisotropy might rotate in the temporal sense to compensate the broken axial asymmetry. In the latter scenario, the asymmetry is interpreted as caused by the largescale flow effect, e.g., the expansion of the solar wind plasma from corona to the heliosphere stretches the wavelengths of turbulence in the radial direction from the sun, while the expansion does not stretch the wavelengths perpendicular to the radial direction.
Coexistence with eddies
The critical balance hypothesis postulates that both Alfvén waves and eddies regulate each other and imply that the both fluctuation types coexist at the same time. It is an important task to study the existence of eddies around the largescale magnetic field direction. Also, the threedimensional picture of plasma turbulence may essentially be different from the twodimensional one due to the additional degree of freedom in the directions around the mean magnetic field. Comparison in direct numerical simulations between twodimensional and threedimensional spatial settings would give a hint on this question.
Spectral tensor
We have treated the energy of the fluctuating field simply as a scalar E, but the fluctuations are found in the plasma quantities (density, flow velocity, temperature), the electric field, and the magnetic field. Furthermore, the fluctuation energy can be analyzed with respect to the field magnitude as well as to the components for the vectorial quantity, e.g., fluctuation parallel or perpendicular to the largemagnetic field. Since the energy spectrum is obtained as the Fourier transform of the correlation function, the analysis of the spectral energy is extended to the analysis of the spectral tensor in the wavevectorfrequency domain. We here give an overview of the spectral tensor analysis for plasma turbulence.
Compressible and incompressible sense
The energy spectrum for the vectorial field is obtained by computing the cross spectral density matrix. Each element of the matrix represents essentially a correlations between different components of the fluctuating field (offdiagonal elements) or for the same component (diagonal elements). One may then compute the trace of the matrix and use it as the total fluctuation energy, as the trace is invariant under the coordinate system rotation. The diagonal and the offdiagonal elements of the tensor can be used in the data analysis by choosing a reasonable coordinate system. For example, the diagonal elements of the matrix in the meanfieldaligned coordinate system (oriented to the largescale magnetic field direction) are the measure of the compressible (parallel to the largescale field direction) and the incompressible sense (perpendicular to the largescale field) of the fluctuating field. Another choice of the coordinate system is the eigenvector system, oriented to the principal axes of the fluctuations. If the fluctuating field is divergencefree, the direction of the minimum fluctuating field can be used as a measure of the wavevector direction with the 180degree ambiguity (because one can determine only the normal direction to the plane of circular or elliptic polarization). Velocity fluctuations longitudinal and transverse to the wavevectors can be evaluated in the tensor analysis.
Field rotation sense
One may also use the offdiagonal elements of the spectral density matrix. These elements contain the information of the field rotation sense. The sense of the field rotation and the eccentricity of elliptically polarizing field can be obtained from the offdiagonal elements (temporal polarization). One may also compute the spatial helical sense of the fluctuation, and the helicity spectrum can be obtained directly in the wavevector domain using multispacecraft measurements. Not only energy but also helicity plays an important role in plasma turbulence, as in a closed system (bounded by a magnetic surface) both the total energy and the total helicity are the conserved quantities in ideal magnetohydrodynamics. Helical sense of the field can be measured for the magnetic field (magnetic helicity) and the flow velocity (kinetic helicity). Also, the energy spectrum can be computed for the righthand and lefthand circularly polarized waves separately, which is a useful tool to study the detailed processes of wave–wave interactions.
Correlation between plasma and magnetic field
Correlation between the flow velocity and the magnetic field can be used to study the cross helicity in plasma turbulence, which is a measure of the difference between Alfvén waves propagating forward to the largescale magnetic field and those propagating backward in incompressible magnetohydrodynamics. However, in the kinetic regime, the concept of the flow velocity breaks down, and one must look in detail at the fluctuation or the disturbance of the velocity distribution function. The use of the energy spectrum for the electric and the magnetic fields is valid in the kinetic regime but the energy spectrum may be different between the electric and the magnetic field, depending on the dispersion effect of the fluctuations. Also, the residual energy between the kinetic and the magnetic ones serves as a useful diagnostics tool in plasma turbulence.
The final process of turbulence (at the highest wavenumbers) after the energy cascade is the energy dissipation. The fluctuation energy is converted into the thermal energy. Different mechanisms exist in wave–particle interactions. The energy is transferred into the thermal one, for example, through the cyclotron resonance between waves and particles by means of the wave electric field perpendicular to the largescale magnetic field or the Landau resonance by means of the parallel electric field. One may also track the time evolution of the thermal energy in plasma turbulence to study how much the medium is heated by turbulence.
Phase information
Turbulence is often associated with the random motion of the fluid. Naively speaking, one may anticipate that the fluctuation statistics exhibits the Gaussian distribution. In fact, we have also used the Gaussian frequency distribution to model the energy spectrum in the wavenumber–frequency domain. Fluctuations following the Gaussian distribution are called selfsimilar, that is, the statistics is scalable to different spatial or time lengths, and the fluctuations on all the possible length scales fill the space (which gives the notion of selfsimilarity).
Turbulence theories, however, predict that the fluctuations statistics should not be strictly Gaussian. One of the important properties of the Gaussian distribution is that the phases of the Fourier transformed fields are randomly distributed. If the distribution were Gaussian, waves comprising the fluctuations are completely incoherent and uncorrelated to each other. For completely randomphase fluctuations, there is no energy transport in the spectral domain, and the picture of the energy cascade in the inertial range breaks down. Deviation from the Gaussian statistics can be found in many physical systems. Fluctuations in the spatial coordinate or in the time series data often exhibit sparsely localized structures or spiky signals, respectively. In other words, fluctuations are not selfsimilar in that the picture of the spatialfilling pattern on all the scales is no longer valid. In fact, in a real turbulent flow, smallscale eddies or fluctuations become increasingly sparse, or intermittent, which is a sign of broken selfsimilarity. Realization of intermittency can be found in coherent wave–wave interactions (that are the main driver of the energy transport in the inertial range) and formation of coherent structures such as current sheets in plasmas. The appearance or the degree of intermittency can be studied using the probability density function (PDF) of the fluctuations. Or one may compute higherorder moments or cumulants of the PDFs and associate the deviation from the Gaussian distribution with the degree of intermittency. Multiple wave couplings can be studied using the method of higherorder correlations. Threefield correlation is called the bispectrum and it is a measure of phase coherence or strength of threewave coupling. Likewise, one may compute fourfield correlation (called the trispectrum) which is a measure of fourwave coupling. Multispacecraft measurements in space can be used for the studies of multiple wave couplings in the wavevectorfrequency domain. The extension of the fourdimensional energy spectrum to the tensorial treatment and the higherorder moments will give a more complete picture of plasma turbulence in space and astrophysical systems.
Notes
Acknowledgements
The author is grateful for stimulating discussions and helpful suggestions by a number of colleagues to improve the quality of the manuscript, in particular KarlHeinz Glassmeier, Tohru Hada, Masahiro Hoshino, Eckart Marsch, and Uwe Motschmann. The author ackowledges financial supports. The work conducted in Graz is supported by Austrian Space Applications Programme at Austrian Research Promotion Agency under contract FFG ASAP12 853994 and Austrian Science Fund (FWF) under the contract P28764N27. The work conducted in Braunschweig is financially supported by German Science Foundation under the contract DFG MO 539/201. The author also acknowledges the hospitality of the University of Tokyo during the research stay supported by Invitational Fellowship for Research in Japan (Shortterm) under the Grant Number FY2017 S17123.
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