Space–time structure and wavevector anisotropy in space plasma turbulence
 3.5k Downloads
 2 Citations
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.
References
 Akhiezer AI, Akhiezer IA, Polovin RV, Sitenko AG, Stepanov KN (1975) Plasma electrodynamics, Vol. 2: Nonlinear theory and fluctuations. Pergamon, New YorkGoogle Scholar
 Alexandrova O, Saur J, Lacombe C, Mangeney A, Mitchell J, Schwartz SJ, Robert P (2009) Universality of solarwind turbulent spectrum from MHD to electron scales. Phys Rev Lett 103:165003. https://doi.org/10.1103/PhysRevLett.103.165003 ADSCrossRefGoogle Scholar
 Bale SD, Kellogg PJ, Mozer FS, Horbury TS, Rème H (2005) Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence. Phys Rev Lett 94:215002. https://doi.org/10.1103/PhysRevLett.94.215002 ADSCrossRefGoogle Scholar
 Battaner E (1996) Astrophysical fluid dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Baumjohann W, Treumann RA, Georgescu E, Haerendel G, Fornaçon KH, Auster HU (1999) Waveform and packet structure of lion roars. Ann Geophys 17:1528–1534. https://doi.org/10.1007/s0058599915289 ADSCrossRefGoogle Scholar
 Baumjohann W, Georgescu E, Fornaçon KH, Auster HU, Treumann RA, Haerendel G (2000) Magnetospheric lion roars. Ann Geophys 18:406–410. https://doi.org/10.1007/s0058500004062 ADSCrossRefGoogle Scholar
 Bieber JW, Matthaeus WH, Smith CW, Wanner W, Kallenrode MB, Wibberenz G (1994) Proton and electron mean free paths: the Palmer consensus revisited. Astrophys J 420:294–306. https://doi.org/10.1086/173559 ADSCrossRefGoogle Scholar
 Bieber JW, Wannger W, Matthaeus WH (1996) Dominant twodimensional solar wind turbulence with implications for cosmic ray transport. J Geophys Res 101:2511–2522. https://doi.org/10.1029/95JA02588 ADSCrossRefGoogle Scholar
 Bingham R, Shapiro VD, Shukla PK, Trines R (2004) Lowerhybrid wave activity and reconnection at the magnetosphere. Phys Scr 113:144–148. https://doi.org/10.1238/Physica.Topical.113a00144 CrossRefGoogle Scholar
 Biskamp D (2003) Magnetohydrodynamic turbulence. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Bruno R, Carbone V (2013) The solar wind as a turbulence laboratory. Living Rev Sol Phys. https://doi.org/10.12942/lrsp20132
 Capon J (1969) Highresolution frequencywavenumber spectrum analysis. Proc IEEE 57:1408–1418. https://doi.org/10.1109/PROC.1969.7278 CrossRefGoogle Scholar
 Carbone V, Malara F, Veltri P (1995) A model for the threedimensional magnetic field correlation spectra of lowfrequency solar wind fluctuations during Alfvénic periods. J Geophys Res 100:1763–1778. https://doi.org/10.1029/94JA02500 ADSCrossRefGoogle Scholar
 Carbone F, SorrisoValvo L, Versace C, Strangi G, Bartolino R (2011) Anisotropy of spatiotemporal decorrelation in electrohydrodynamic turbulence. Phys Rev Lett 106:114502. https://doi.org/10.1103/PhysRevLett.106.114502 ADSCrossRefGoogle Scholar
 Cardinali A, Castaldo C, Cesario R, De Marco F, Paoletti F (2002) Analysis of the heating scenarios of the ion Bernstein wave (IBW) experiment in Frascati tokamak upgrade. Nucl Fus 42:427–440. https://doi.org/10.1088/00295515/42/4/307 ADSCrossRefGoogle Scholar
 Cerri SS, Califano F, Jenko F, Told D, Rincon F (2016) Subprotonscale cascades in solar wind turbulence: driven hybridkinetic simulations. Astrophys J Lett 822:L12. https://doi.org/10.3847/20418205/822/1/L12 ADSCrossRefGoogle Scholar
 Chandran BDG (2008) Strong anisotropic MHD turbulence with cross helicity. Astrophys J 685:646–658. https://doi.org/10.1086/589432 ADSCrossRefGoogle Scholar
 Chang O, Gary SP, Wang J (2013) Whistler turbulence at variable electron beta: threedimensional particleincell simulations. J Geophys Res 118:2824–2833. https://doi.org/10.1002/jgra.50365 CrossRefGoogle Scholar
 Chen CHK, Mallet A, Schekochihin AA, Horbury TS, Wicks RT, Bale SD (2012) Threedimensional structure of solar wind turbulence. Astrophys J 758:120. https://doi.org/10.1088/0004637X/758/2/120 ADSCrossRefGoogle Scholar
 Chen CHK, Boldyrev S, Xia Q, Perez JC (2013) Nature of subproton scale turbulence in the solar wind. Phys Rev Lett 110:225002. https://doi.org/10.1103/PhysRevLett.110.225002 ADSCrossRefGoogle Scholar
 Cho J, Vishnian ET (2000) The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys J 539:273–282. https://doi.org/10.1086/309213 ADSCrossRefGoogle Scholar
 Coleman JPJ (1968) Turbulence, viscosity, and dissipation in the solarwind plasma. Astrophys J 153:371–388. https://doi.org/10.1086/149674 ADSCrossRefGoogle Scholar
 Comişel H, Verscharen D, Narita Y, Motschmann U (2013) Spectral evolution of twodimensional kinetic plasma turbulence in the wavenumber–frequency domain. Phys Plasmas 20:090701. https://doi.org/10.1063/1.4820936 ADSCrossRefGoogle Scholar
 Corrsin S (1963) Estimates of the relations between Eulerian and Lagrangian scales in large Reynolds number turbulence. J Atmos Sci 20:115–119. https://doi.org/10.1175/15200469(1963)020%3c0115:EOTRBE%3e2.0.CO;2
 Dasso S, Milano LJ, Matthaeus WH, Smith CW (2005) Anisotropy in fast and slow solar wind fluctuations. Astrophys J 635:L181–L184. https://doi.org/10.1086/499559 ADSCrossRefGoogle Scholar
 Derby JNF (1978) Modulational instability of finiteamplitude, circularly polarized Alfvén waves. Astrophys J 224:1013–1016. https://doi.org/10.1086/156451 ADSCrossRefGoogle Scholar
 Dudok de Wit T, Krasnosel’skikh VV, Bale SD, Dunlop MW, Lühr H, Schwartz SJ, Woolliscroft LJC (1995) Determination of dispersion relations in quasistationary plasma turbulence using dual satellite data. Geophys Res Lett 22:2653–2656. https://doi.org/10.1029/95GL02543 ADSCrossRefGoogle Scholar
 Eastwood JP, Phan TD, Bale SD, Tjulin A (2009) Observations of turbulence generated by magnetic reconnection. Phys Rev Lett 102:035001. https://doi.org/10.1103/PhysRevLett.102.035001 ADSCrossRefGoogle Scholar
 Falco RE (1977) Coherent motions in the outer region of turbulent boundary layers. Phys Fluids 20:S124–S132. https://doi.org/10.1063/1.861721 ADSCrossRefGoogle Scholar
 Forman MA, Wicks RT, Horbury TS (2011) Detailed fit of “critical balance” theory to solar wind turbulence measurements. Astrophys J 733:76. https://doi.org/10.1088/0004637X/733/2/76 ADSCrossRefGoogle Scholar
 Frisch U (1995) Turbulence: the legacy of A. N. Kolmogorov. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Gary SP (1986) Lowfrequency waves in a highbeta collisionless plasma: polarization, compressibility and helicity. J Plasma Phys 35:431–447. https://doi.org/10.1017/S0022377800011442 ADSCrossRefGoogle Scholar
 Gary SP (1993) Theory of space plasma microinstabilities. Cambridge Atmospheric and Space Science Series. Cambridge University Press, CambridgeGoogle Scholar
 Gary SP (2013) Test for wavevector anisotropy in plasma turbulence cascades. Astrophys J 769:36. https://doi.org/10.1088/0004637X/769/1/36 ADSCrossRefGoogle Scholar
 Gary SP, Chang O, Wang J (2012) Forward cascade of whistler turbulence: threedimensional particleincell simulations. Astrophys J 755:142. https://doi.org/10.1088/0004637X/755/2/142 ADSCrossRefGoogle Scholar
 Glassmeier KH, Motschmann U, Dunlop M, Balogh A, Acuña MH, Carr C, Musmann G, Fornaçon KH, Schweda K, Vogt J, Georgescu E, Buchert S (2001) Cluster as a wave telescope—first results from the fluxgate magnetometer. Ann Geophys 19:1439–1447. https://doi.org/10.5194/angeo1914392001 ADSCrossRefGoogle Scholar
 Goldreich P, Sridhar S (1995) Toward a theory of interstellar turbulence. II. Strong Alfvénic turbulence. Astrophys J 438:763–775. https://doi.org/10.1086/175121 ADSCrossRefGoogle Scholar
 Goldstein ML (1978) An instability of finite amplitude circularly polarized Alfvén waves. Astrophys J 219:700–704. https://doi.org/10.1086/155829 ADSMathSciNetCrossRefGoogle Scholar
 Graham DB, Khotyaintsev YV, Norgren C, Vaivads A, André M, ToledoRedondo S, PA L, Marklund GT, Ergun WR R E Paterson, Gershman DJ, Giles BL, Pollock CJ, Dorelli JC, Avanov LA, Lavraud B, Saito Y, Magnes W, Russell CT, Strangeway RJ, Torbet RB, Burch JL (2017) Lower hybrid waves in the ion diffusion and magnetospheric inflow regions. J Geophys Res 122:517–533. https://doi.org/10.1002/2016JA023572 CrossRefGoogle Scholar
 He GW, Zhang JB (2006) Elliptic model for space–time correlations in turbulent shear flows. Phys Rev E 73:055303(R). https://doi.org/10.1103/PhysRevE.73.055303 ADSCrossRefGoogle Scholar
 He J, Tu C, Marsch E, Bourouaine S, Pei Z (2013) Radial evolution of the wavevector anisotropy of solar wind turbulence between 0.3 and 1 AU. Astrophys J 773:72. https://doi.org/10.1088/0004637X/773/1/72 ADSCrossRefGoogle Scholar
 Hollweg JV (1999) Kinetic Alfvén wave revisited. J Geophys Res 104:14,811–14,819. https://doi.org/10.1029/1998JA900132 ADSCrossRefGoogle Scholar
 Hoppe MM, Russell CT (1983) Plasma rest frame frequencies and polarizations of the lowfrequency upstream waves: ISEE 1 and 2 observations. J Geophys Res 88:2021–2028. https://doi.org/10.1029/JA088iA03p02021 ADSCrossRefGoogle Scholar
 Horbury TS, Forman M, Oughton S (2008) Anisotropic scaling of magnetohydrodynamic turbulence. Phys Rev Lett 101:175005. https://doi.org/10.1103/PhysRevLett.101.175005 ADSCrossRefGoogle Scholar
 Howes GG, Tenbarge JM, Dorland W, Quataert E, Schekochihin AA, Numata R, Tatsuno T (2011) Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys Rev Lett 107:035004. https://doi.org/10.1103/PhysRevLett.107.035004 ADSCrossRefGoogle Scholar
 Howes GG, Bale SD, Klein KG, Chen CHK, Salem CS, TenBarge JM (2012) The slowmode nature of compressible wave power in solar wind turbulence. Astrophys J Lett 753:L19. https://doi.org/10.1088/20418205/753/1/L19 ADSCrossRefGoogle Scholar
 Inoue E (1951) On the turbulent diffusion in the atmosphere. II. J Met Soc Jpn 29:246–252. https://doi.org/10.2151/jmsj1923.29.7_246 CrossRefGoogle Scholar
 Intrator T, Myra JR, D’Ippolit D (2003) Threedimensional finiteelement model of the ion Bernstein wave antenna and excitation of coaxial electrostatic edge modes in the tokamak fusion test reactor. Nucl Fus 43:531–538. https://doi.org/10.1088/00295515/43/7/304 ADSCrossRefGoogle Scholar
 Iroshnikov PS (1964) Turbulence of a conducting fluid in a strong magnetic field. Sov Astron 7:566ADSMathSciNetGoogle Scholar
 Jenkins TG, Austin TV, Smithe DN, Loverich J, Hakim AH (2013) Timedomain simulation of nonlinear radiofrequency phenomena. Phys Plasmas 20:012116. https://doi.org/10.1063/1.4776704 ADSCrossRefGoogle Scholar
 Kadomtsev BB (1960) Convective pinch instability. Sov Phys JETP 10:780MathSciNetGoogle Scholar
 Katoh Y (2014) A simulation study of the propagation of whistlermode chorus in the Earth’s inner magnetosphere. Earth Planets Space 66:6. https://doi.org/10.1186/18805981666 ADSCrossRefGoogle Scholar
 Kiyani KH, Chapman SC, Sahraoui F, Hnat B, Fauvarque O, Khotyaintsev YV (2013) Enhanced magnetic compressibility and isotropic scale invariance at subion Larmor scales in solar wind turbulence. Astrophys J 763:10. https://doi.org/10.1088/0004637X/763/1/10 ADSCrossRefGoogle Scholar
 Klein KG, Howes GG, TenBarge JM, Bale SD, Chen CHK, Salem CS (2012) Using synthetic spacecraft data to interpret compressible fluctuations in solar wind turbulence. Astrophys J 755:159. https://doi.org/10.1088/0004637X/755/2/159 ADSCrossRefGoogle Scholar
 Kolmogorov AN (1991) The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Proc R Soc A 434:9–13. https://doi.org/10.1098/rspa.1991.0075 first published in Russian in Dokl. Akad. Nauk. SSSR, 30 (1941) 299–303ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Korsholm SB, Stejner M, Bindslev H, Furtula V, Leipold F, Meo F, Michelsen PK, Moseev D, Nielsen SK, Salewski M, de Baar M, Delabie E, Kantor M, Bürger A, Team T (2011) Measurements of intrinsic ion Bernstein waves in a tokamak by collective Thomson scattering. Phys Rev Lett 106:165004. https://doi.org/10.1103/PhysRevLett.106.165004 ADSCrossRefGoogle Scholar
 Kraichnan RH (1964) Kolmogorov’s hypothesis and Eulerian turbulence theory. Phys Fluids 7:1723–1734. https://doi.org/10.1063/1.2746572 ADSMathSciNetCrossRefzbMATHGoogle Scholar
 Kraichnan R (1965a) Inertialrange spectrum of hydromagnetic turbulence. Phys Fluids 8:1385–1387. https://doi.org/10.1063/1.1761412 ADSCrossRefGoogle Scholar
 Kraichnan R (1965b) Lagrangianhistory closure approximation for turbulence. Phys Fluids 8:575–598. https://doi.org/10.1063/1.1761271 ADSMathSciNetCrossRefGoogle Scholar
 Leamon RJ, Smith CW, Ness NF, Matthaeus WH, Wong HK (1998) Observational constraints on the dynamics of the interplanetary magnetic field dissipation range. J Geophys Res 103:4775–4787. https://doi.org/10.1029/97JA03394 ADSCrossRefGoogle Scholar
 Lefeuvre F, Parrot M, Neubert T (1982) Wave normal directions and wave distribution functions for groundbased transmitter signals observed on GEOS 1. J Geophys Res 87:6203–6217. https://doi.org/10.1029/JA087iA08p06203 ADSCrossRefGoogle Scholar
 Longtin M, Sonnerup B (1986) Modulational instability of circularly polarized Alfvén waves. J Geophys Res 91:798–801. https://doi.org/10.1029/JA091iA06p06816 CrossRefGoogle Scholar
 Lysak RL (2008) On the dispersion relation for the kinetic Alfvén wave in an inhomogeneous plasma. Phys Plasmas 15:062901. https://doi.org/10.1063/1.2918742 ADSCrossRefGoogle Scholar
 Lysak RL, Lotko W (1996) On the kinetic dispersion relation for shear Alfvén waves. J Geophys Res 101:5085–5094. https://doi.org/10.1029/95JA03712 ADSCrossRefGoogle Scholar
 Maneva YG, Viñas AF, Ofman L (2013) Turbulent heating and acceleration of He\(^{++}\) ions by spectra of Alfvéncyclotron waves in the expanding solar wind: 1.5D hybrid simulations. J Geophys Res 118:2842–2853. https://doi.org/10.1002/jgra.50363 CrossRefGoogle Scholar
 Maneva YG, Viñas AF, Moya PS, Wicks RT, Poedts S (2015) Dissipation of parallel and oblique Alfvéncyclotron waves: implications for heating of alpha particles in the solar wind. Astrophys J 814:33. https://doi.org/10.1088/0004637X/814/1/33 ADSCrossRefGoogle Scholar
 Marsch E, Chang T (1982) Lower hybrid waves in the solar wind. Geophys Res Lett 9:1155–1158. https://doi.org/10.1029/GL009i010p01155 ADSCrossRefGoogle Scholar
 Marsch E, Chang T (1983) Electromagnetic lower hybrid waves in the solar wind. J Geophys Res 88:6869–6880. https://doi.org/10.1029/JA088iA09p06869 ADSCrossRefGoogle Scholar
 Marsch E, Tu CY (1990) On the radial evolution of MHD turbulence in the inner heliosphere. J Geophys Res 95:8211–8229. https://doi.org/10.1029/JA095iA06p08211 ADSCrossRefGoogle Scholar
 Marsch E, Tu CY (2001) Evidence for pitch angle diffusion of solar wind protons in resonance with cyclotron waves. J Geophys Res 106:8357–8361. https://doi.org/10.1029/2000JA000414 ADSCrossRefGoogle Scholar
 Matthaeus WH, Goldstein ML (1982) Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J Geophys Res 87:6011–6028. https://doi.org/10.1029/JA087iA08p06011 ADSCrossRefGoogle Scholar
 Matthaeus WH, Ghosh S (1999) Spectral decomposition of solar wind turbulence: threecomponent model. In: Habbal SR, Esser R, Vollweg JV, Isenberg PA (eds) The Solar Wind Nine Conference, AIP Conference Proceedings, vol 471, pp 519–522. https://doi.org/10.1063/1.58688
 Matthaeus WH, Goldstein ML, Roberts DA (1990) Evidence for the presence of quasitwodimensional nearly incompressible fluctuations in the solar wind. J Geophys Res 95:20673–20683. https://doi.org/10.1029/JA095iA12p20673 ADSCrossRefGoogle Scholar
 Matthaeus WH, Gosh S, Ougton S, Roberts DA (1996) Anisotropic threedimensional MHD turbulence. J Geophys Res 101:7619–7630. https://doi.org/10.1029/95JA03830 ADSCrossRefGoogle Scholar
 Mio K, Ogino T, Minami K, Takeda S (1976) Modulational instability and envelopesolitons for nonlinear Alfvén waves propagating along the magnetic field in plasmas. J Phys Soc Jpn 41:667–673. https://doi.org/10.1143/JPSJ.41.667 ADSCrossRefzbMATHGoogle Scholar
 Mjølhus E (1976) On the modulational instability of hydromagnetic waves parallel to the magnetic field. J Plasma Phys 16:321–334. https://doi.org/10.1017/S0022377800020249 ADSCrossRefGoogle Scholar
 Motschmann U, Woodward TI, Glassmeier KH, Southwood DJ, Pinçon JL (1996) Wavelength and direction filtering by magnetic measurements at satellite arrays: generalized minimum variance analysis. J Geophys Res 101:4961–4966. https://doi.org/10.1029/95JA03471 ADSCrossRefGoogle Scholar
 Narita Y (2014) Fourdimensional energy spectrum for space–time structure of plasma turbulence. Nonlinear Process Geophys 21:41–47. https://doi.org/10.5194/npg21412014 ADSCrossRefGoogle Scholar
 Narita Y (2015) Nonelliptic wavevector anisotropy for magnetohydrodynamic turbulence. Ann Geophys 33:1413–1419. https://doi.org/10.5194/angeo3314132015 ADSCrossRefGoogle Scholar
 Narita Y (2017a) Error estimate of Taylor’s frozenin flow hypothesis in the spectral domain. Ann Geophys 35:325–331. https://doi.org/10.5194/angeo353252017 ADSCrossRefGoogle Scholar
 Narita Y (2017b) Review article: Wave analysis methods for space plasma experiment. Nonlinear Process Geophys 24:203–214. https://doi.org/10.5194/npg242032017 ADSCrossRefGoogle Scholar
 Narita Y, Gary SP, Saito S, Glassmeier KH, Motschmann U (2011a) Dispersion relation analysis of solar wind turbulence. Geophys Res Lett 38:L05101. https://doi.org/10.1029/2010GL046588 ADSCrossRefGoogle Scholar
 Narita Y, Glassmeier KH, Motschmann U (2011b) Highresolution wave number spectrum using multipoint measurements in space—the multipoint signal resonator (MSR) technique. Ann Geophys 29:351–360. https://doi.org/10.5194/angeo293512011 ADSCrossRefGoogle Scholar
 Narita Y, Comişel H, Motschmann U (2014) Spatial structure of ionscale plasma turbulence. Front Phys 2:13. https://doi.org/10.3389/fphy.2014.00013 CrossRefGoogle Scholar
 Narita Y, Nakamura R, Baumjohann W, Glassmeier KH, Motschmann U, Giles B, Magnes W, Fischer D, Torbert RB, Russell CT, Strangeway RJ, Burch JL, Nariyuki Y, Saito S, Gary SP (2016) On electronscale whistler turbulence in the solar wind. Astrophys J Lett 827:L8. https://doi.org/10.3847/20418205/827/1/L8 ADSCrossRefGoogle Scholar
 Nariyuki Y, Hada T (2006) Remarks on nonlinear relation among phases and frequencies in modulational instabilities of parallel propagating Alfvén waves. Nonlinear Process Geophys 13:425–441. https://doi.org/10.5194/npg134252006 ADSCrossRefGoogle Scholar
 Neubauer FM (1990) Use of an array of satellites as a wave telescope. J Geophys Res 95:19,115–19,122. https://doi.org/10.1029/JA095iA11p19115 ADSCrossRefGoogle Scholar
 Nomura R, Shiokawa K, Sakaguchi K, Otsuka Y, Connors M (2012) Polarization of Pc1/EMIC waves and related proton auroras observed at subauroral latitudes. J Geophys Res 117:A02318. https://doi.org/10.1029/2011JA017241 ADSCrossRefGoogle Scholar
 Ofman L, Viñas AF, Maneva Y (2014) Twodimensional hybrid models of H\(^{+}\)He\(^{++}\) expanding solar wind plasma heating. J Geophys Res 119:4223–4238. https://doi.org/10.1002/2013JA019590 CrossRefGoogle Scholar
 Oscarsson T (1994) Dual principles in maximum entropy reconstruction of the wave distribution function. J Comput Phys 110:221–233. https://doi.org/10.1006/jcph.1994.1020 ADSCrossRefzbMATHGoogle Scholar
 Oscarsson TE, Rönnmark KG (1989) Reconstruction of wave distribution functions in warm plasmas. J Geophys Res 94:2417–2428. https://doi.org/10.1029/JA094iA03p02417 ADSCrossRefGoogle Scholar
 Oscarsson TE, Rönnmark KG (1990) A combined wave distribution function and stability analysis of viking particle and lowfrequency wave data. J Geophys Res 95:21,187–21,202. https://doi.org/10.1029/JA095iA12p21187 ADSCrossRefGoogle Scholar
 Oscarsson T, Stenberg G, Santolik O (2001) Wave mode identification via wave distribution function analysis. Phys Chem Earth C 26:229–235. https://doi.org/10.1016/S14641917(00)001136 CrossRefGoogle Scholar
 Osman KT, Horbury TS (2009) Multispacecraft measurement of anisotropic power levels and scaling in solar wind turbulence. Ann Geophys 27:3019–3025. https://doi.org/10.5194/angeo2730192009 ADSCrossRefGoogle Scholar
 Ozak N, Ofman L, Viñas AF (2015) Ion heating in inhomogeneous expanding solar wind plasma: the role of parallel and oblique ion–cyclotron waves. Astrophys J 799:77. https://doi.org/10.1088/0004637X/799/1/77 ADSCrossRefGoogle Scholar
 Perschke C, Narita Y, Gary SP, Motschmann U, Glassmeier KH (2013) Dispersion relation analysis of turulent magnetic field fluctuations in fast solar wind. Ann Geophys 31:1949–1955. https://doi.org/10.5194/angeo3119492013 ADSCrossRefGoogle Scholar
 Perschke C, Motschmann Narita U, Glassmeier KH (2014) Multispacecraft observations linear modes and sideband waves in ionscale solar wind turbulence. Astrophys J 793:L25. https://doi.org/10.1088/20418205/793/2/L25 ADSCrossRefGoogle Scholar
 Perschke C, Narita Y, Motschmann U, Glassmeier KH (2016) Observational test for a random sweeping model in solar wind turbulence. Phys Rev Lett 116:125101. https://doi.org/10.1103/PhysRevLett.116.125101 ADSCrossRefGoogle Scholar
 Petrosyan A, Balogh A, Goldstein ML, Léorat J, Marsch E, Petrovay K, Roberts B, von Steiger R, Vial JC (2010) Turbulence in the solar atmosphere and solar wind. Space Sci Rev 156:135–238. https://doi.org/10.1007/s1121401096943 ADSCrossRefGoogle Scholar
 Pinçon JL, Lefeuvre F (1991) Local characterization of homogeneous turbulence in a space plasma from simultaneous measurement of field components at several points in space. J Geophys Res 96:1789–1802. https://doi.org/10.1029/90JA02183 ADSCrossRefGoogle Scholar
 Podesta JJ, Roberts DA, Goldstein ML (2007) Spectral exponents of kinetic and magnetic energy spectra in solar wind turbulence. Astrophys J 664:543–548. https://doi.org/10.1086/519211 ADSCrossRefGoogle Scholar
 Richardson LF (1926) Atmospheric diffusion shown on a distance–neighbour graph. Proc R Soc Lond A 110:709–737. https://doi.org/10.1098/rspa.1926.0043 ADSCrossRefGoogle Scholar
 Roberts OW, Li X, Li B (2013) Kinetic plasma turbulence in the fast solar wind measured by Cluster. Astrophys J 769:58. https://doi.org/10.1088/0004637X/769/1/58 ADSCrossRefGoogle Scholar
 Roberts OW, Li X, Jeska L (2015a) Evidence of the ion cyclotron resonance at proton kinetic scales in the solar wind. Astrophys J 802:1. https://doi.org/10.1088/0004637X/802/1/1 ADSCrossRefGoogle Scholar
 Roberts OW, Li X, Jeska L (2015b) A statistical study of the solar wind turbulence at ion kinetic scales using the kfiltering technique and Cluster data. Astrophys J 802:2. https://doi.org/10.1088/0004637X/802/1/2 ADSCrossRefGoogle Scholar
 Saddoughi SG, Veeravalli SV (1994) Local isotropy in turbulent boundary layers at high Reynolds number. J Fluid Mech 268:333–372. https://doi.org/10.1017/S0022112094001370 ADSCrossRefGoogle Scholar
 Sahraoui F, Goldstein ML, Robert P, Khotyaintsev YV (2009) Evidence of a cascade and dissipation of solarwind turbulence at the electron gyroscale. Phys Rev Lett 102:231102. https://doi.org/10.1103/PhysRevLett.102.231102 ADSCrossRefGoogle Scholar
 Sahraoui F, Goldstein ML, Belmont G, Canu P, Rezeau L (2010) Three dimensional anisotropic \(k\) spectra of turbulence at subproton scales in the solar wind. Phys Rev Lett 105:131101. https://doi.org/10.1103/PhysRevLett.105.131101 ADSCrossRefGoogle Scholar
 Saito S, Gary SP, Li H, Narita Y (2008) Whistler turbulence: particleincell simulations. Phys Plasmas 15:102305. https://doi.org/10.1063/1.2997339 ADSCrossRefGoogle Scholar
 Salem CS, Howes GG, Sundkvist D, Bale SD, Chaston CC, Chen CHK, Mozer FS (2012) Identification of kinetic Alfvén wave turbulence in the solar wind. Astrophys J Lett 745:L9. https://doi.org/10.1088/20418205/745/1/L9 ADSCrossRefGoogle Scholar
 Santolik O, Parrot M (1996) The wave distribution function in a hot magnetospheric plasma: the direct problem. J Geophys Res 101:10,639–10,652. https://doi.org/10.1029/95JA03510 ADSCrossRefGoogle Scholar
 Saur J, Bieber JW (1999) Geometry of lowfrequency solar wind magnetic turbulence: evidence for radially aligned Alfvénic fluctuations. J Geophys Res 104:9975–9988. https://doi.org/10.1029/1998JA900077 ADSCrossRefGoogle Scholar
 Schmidt RO (1986) Multiple emitter location and signal parameter estimation. IEEE Trans Ant Propag 34:276–280. https://doi.org/10.1109/TAP.1986.1143830 ADSCrossRefGoogle Scholar
 Schwenn R, Marsch E (1991) Physics of the inner heliosphere II: particles, waves, and turbulence. Springer, Heidelberg. https://doi.org/10.1007/9783642753640 CrossRefGoogle Scholar
 Servidio S, Carbone V, Dmitruk P, Matthaeus WH (2011) Time decorrelation in isotropic magnetohydrodynamic turbulence. Europhys Lett 96:55003. https://doi.org/10.1209/02955075/96/55003 ADSCrossRefGoogle Scholar
 Shebalin JV, Matthaeus WH, Montgomery D (1983) Anisotropy in MHD turbulence due to a mean magnetic field. J Plasma Phys 29:525–547. https://doi.org/10.1017/S0022377800000933 ADSCrossRefGoogle Scholar
 Stansby D, Horbury TS, Chen CHK, Matteini L (2016) Experimental determination of whistler wave dispersion relation in the solar wind. Astrophys J Lett 829:L16. https://doi.org/10.3847/20418205/829/1/L16 ADSCrossRefGoogle Scholar
 Stix TH (1992) Waves in Plasmas. Springer, New YorkGoogle Scholar
 Storey LRO, Lefeuvre F (1979) The analysis of 6component measurements of a random electromagnetic wave field in a magnetoplasma—I. The direct problem. Geophys J R Astron Soc 56:255–269. https://doi.org/10.1111/j.1365246X.1979.tb00163.x ADSCrossRefGoogle Scholar
 Storey LRO, Lefeuvre F (1980) The analysis of 6component measurements of a random electromagnetic wave field in a magnetoplasma—II. The integration kernels. Geophys J R Astron Soc 62:173–194. https://doi.org/10.1111/j.1365246X.1980.tb04850.x ADSCrossRefGoogle Scholar
 Taylor GI (1938) The spectrum of turbulence. Proc R Soc Lond A 164:476–490. https://doi.org/10.1098/rspa.1938.0032 ADSCrossRefzbMATHGoogle Scholar
 TenBarge JM, Podesta JJ, Klein KG, Howes GG (2012) Interpreting magnetic variance anisotropy measurements in the solar wind. Astrophys J 753:107. https://doi.org/10.1088/0004637X/753/2/107 ADSCrossRefGoogle Scholar
 Tennekes H (1975) Eulerian and Lagrangian time microscales in isotropic turbulence. J Fluid Mech 67:561–567. https://doi.org/10.1017/S0022112075000468 ADSCrossRefzbMATHGoogle Scholar
 Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, BostonzbMATHGoogle Scholar
 Terasawa T, Hoshino M, Sakai JI, Hada T (1986) Decay instability of finiteamplitude circularly polarized Alfvén waves: a numerical simulation of stimulated brillouin scattering. J Geophys Res 91:4171–4187. https://doi.org/10.1029/JA091iA04p04171 ADSCrossRefGoogle Scholar
 Toida M, Suzuki T, Ohsawa Y (2004) Collective behavior of ion Bernstein waves in a multiionspecies plasma. Phys Plasmas 11:3028–3034. https://doi.org/10.1063/1.1712977 ADSCrossRefGoogle Scholar
 Treumann RA, Baumjohann W (1997) Advanced space plasma physics. Imperial College Press, LondonCrossRefGoogle Scholar
 Tsurutani BT, Smith EJ (1984) Magnetosonic waves adjacent to the plasma sheet in the distant magnetotail: ISEE3. Geophys Res Lett 11:331–334. https://doi.org/10.1029/GL011i004p00331 ADSCrossRefGoogle Scholar
 Tsurutani BT, Richardson IG, Thorne RM, Butler W, Smith EJ, Cowley SWH, Gary SP, Akasofu SI, Zwickl RD (1985) Observations of the righthand resonant ion beam instability in the distant plasma sheet boundary layer. J Geophys Res 90:12,159–12,172. https://doi.org/10.1029/JA090iA12p12159 ADSCrossRefGoogle Scholar
 Tsytovich VN (1970) Nonlinear effects in plasma. Plenum Press, New YorkCrossRefGoogle Scholar
 Tu CY, Marsch E (1995) MHD structures, waves and turbulence in the solar wind: observations and theories. Space Sci Rev 73:1–210. https://doi.org/10.1007/BF00748891 ADSCrossRefGoogle Scholar
 Turner AJ, Gogoberidze G, Chapman SC, Hnat B, Müller WC (2011) Nonaxisymmetric anisotropy of solar wind turbulence. Phys Rev Lett 107:095002. https://doi.org/10.1103/PhysRevLett.107.095002
 Vaivads A, André M, Buchert SC, Wahlund JE, Fazakerley AN, CornilleauWehrlin N (2004) Cluster observations of lower hybrid turbulence within thin layers at the magnetopause. Geophys Res Lett 31:L03804. https://doi.org/10.1029/2003GL018142 ADSCrossRefGoogle Scholar
 Vaivads A, Santolik O, Stenberg G, André M, Owen CJ, Canu P, Dunlop M (2007) Source of whistler emissions at the dayside magnetopause. Geophys Res Lett 34:L09106. https://doi.org/10.1029/2006GL029195 ADSCrossRefGoogle Scholar
 Valentini F, Califano F, Veltri P (2010) Twodimensional kinetic turbulence in the solar wind. Phys Rev Lett 104:205002. https://doi.org/10.1103/PhysRevLett.104.205002 ADSCrossRefGoogle Scholar
 Verscharen D, Marsch E, Motschmann U, Müller J (2012) Kinetic cascade beyond magnetohydrodynamics of solar wind turbulence in twodimensional hybrid simulations. Phys Plasmas 19:022305. https://doi.org/10.1063/1.3682960 ADSCrossRefGoogle Scholar
 Warhaft Z (2002) Turbulence in nature and in the laboratory. Proc Natl Acad Sci USA 99:2481–2486. https://doi.org/10.1073/pnas.012580299 ADSCrossRefGoogle Scholar
 Wilczek M, Narita Y (2012) Wavenumberfrequency spectrum for turbulence from a random sweeping hypothesis with mean flow. Phys Rev E 86:066308. https://doi.org/10.1103/PhysRevE.86.066308 ADSCrossRefGoogle Scholar
 Wong HK, Goldstein ML (1986) Parametric instabilities of circularly polarized Alfvén waves including dispersion. J Geophys Res 91:5617–5628. https://doi.org/10.1029/JA091iA05p05617 ADSCrossRefGoogle Scholar
 Yao S, He JS, Tu CY, Wang LH, Marsch E (2013) Smallscale pressurebalanced structures driven by oblique slow mode waves measured in the solar wind. Astrophys J 774:59. https://doi.org/10.1088/0004637X/774/1/59 ADSCrossRefGoogle Scholar
 Yokoi N (2006) Modeling of the turbulent magnetohydrodynamic residualenergy equation using a statistical theory. Phys Plasmas 13:062306. https://doi.org/10.1063/1.2209232 ADSMathSciNetCrossRefGoogle Scholar
 Yokoi N, Hamba F (2007) An application of the turbulent magnetohydrodynamic residualenergy equation model to the solar wind. Phys Plasmas 14:112904. https://doi.org/10.1063/1.2792337 ADSCrossRefGoogle Scholar
 Yokoi N, Rubinstein R, Yoshizawa A, Hamba F (2008) A turbulence model for magnetohydrodynamic plasmas. J Turbul 9:1–25. https://doi.org/10.1080/14685240802433057 MathSciNetCrossRefGoogle Scholar
 Zhang XJ, Zhao YP, Wan BN, Gong XZ, Lin Y, Zhang WY, Mao YZ, Qin CM, Yuan S, Deng X, Wang L, Ju SQ, Chen Y, Li YD, Li JG, Noterdaeme JM, Wukitch SJ (2012) Experimental observation of ion heating by modeconverted ion Bernstein waves in tokamak plasmas. Nucl Fus 52:082003. https://doi.org/10.1088/00295515/52/8/082003 ADSCrossRefGoogle Scholar
 Zhao JS, Voitenko Y, Yu MY, Lu JY, Wu DJ (2014) Properties of shortwavelength oblique Alfvén and slow waves. Astrophys J 793:107. https://doi.org/10.1088/0004637X/793/2/107 ADSCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.