Ion-cyclotron mode is smoothly connected from the MHD Alfvén mode with a left-hand temporal field rotation sense around the mean magnetic field. The frequencies of the ion-cyclotron mode approach the resonance at the ion gyro-frequency for the parallel propagation. The resonance frequency becomes lower for oblique propagations or for a larger value of beta. The electromagnetic ion-cyclotron mode can propagate along the magnetic field line and deposit the energy over a large distance. For example, 1-Hz ion-cyclotron waves transfer the energy from the radiation belt down to the Earth ionosphere and cause pulsating proton aurora [8]. There is evidence for the ion-cyclotron mode in the solar wind (inner heliosphere) [9]. The velocity distribution functions of ions show a peak at the apparent phase speed for the ion-cyclotron mode in the parallel direction to the mean magnetic field and an arc shape centered at the apparent phase speed for the ion-cyclotron mode, a sign of the pitch angle scattering by the ion-cyclotron mode.

Numerical simulations indicate that the ion-cyclotron mode can heat and accelerate the helium alpha particles, as well [10–13].

Kinetic Alfvén mode is obtained as a quasi-perpendicular limit of the ion-cyclotron mode and hence a small-wavelength extension of the MHD Alfvén mode. At a propagation angle of about 70 degrees, the sense of the dispersion relation or the curvature of the dispersion branch becomes flipped from a converging sense of the frequency increase toward the ion-cyclotron resonance frequency into a diverging sense of the frequency increase such as that of the whistler mode. The kinetic Alfvén mode has very low frequencies (below the ion gyro-frequency) and are only moderately damped, and is considered as one of the most relevant or dominant mode in the solar wind. In fact, a number of spacecraft observations are favorably interpreted as realization of the kinetic Alfvén mode in a frequency range between 0.1 and 100 Hz in the spacecraft frame: from the multi-spacecraft k-filtering technique [14, 15] and by other techniques [16–20]. Recent numerical simulations for forced hybrid kinetic turbulence suggest that the kinetic Alfvén mode dominates ion-scale turbulence in a high-beta plasma, and the whistler mode in a low-beta plasma [21].

Kinetic slow mode is a small-wavelength extension of the MHD slow mode and has as low frequencies as that of the kinetic Alfvén mode (depending on the value of beta). While the slow mode is generally considered as a strongly Landau-damped mode, the slow mode acquires only a moderate damping rate for nearly perpendicular propagations. Compressive fluctuations in the solar wind may be slow mode waves [22, 23] or pressure-balanced structures between the magnetic field and the plasma in the solar wind [24, 25].

The whistler mode is smoothly connected from the MHD fast mode with a right-hand field rotation sense around the mean magnetic field [26]. The existence of the whistler mode is indicated in various regions in space plasmas, e.g., as magnetospheric chorus [27], magnetotail right-hand waves [28, 29], magnetosheath or magnetospheric lion roar waves [30, 31], magnetopause-waves [32], and waves departing from magnetic reconnection [33]. In the ion-kinetic domain, the frequencies of the whistler mode are increasingly higher at larger wavenumbers. The dispersion branch begins to split into resonance branches at the ion gyro-frequency and its harmonics as the ion Bernstein mode. The frequencies between different Bernstein modes still retain the dispersion branch for the whistler mode In a low-beta plasma, the quasi-perpendicular whistler mode is more smoothly connected from lower frequencies to higher frequencies. In a high-beta plasma, in contrast, the whistler mode branch almost vanishes because of the resonant splitting into the ion Bernstein mode. Several cases of the solar wind observations are interpreted as the whistler mode on the ion-kinetic scales [34] and the electron-kinetic scales [35, 36].

The ion Bernstein mode is a set of ion resonance branches that appear as a break-up of the whistler branch at the ion gyro-frequency (fundamental mode) as well as the harmonics (second order and higher order). The Bernstein mode can be both electrostatic and electromagnetic, and is particularly suited for transferring the fluctuation energy toward higher frequencies by three-wave couplings [37]. The ion Bernstein mode is extensively studied in Tokamak, fusion, and laboratory plasmas [38–42]. Several observational studies indicate the ion Bernstein mode in the solar wind [43, 44].

Small-amplitude fluctuations in the plasma can develop into turbulence by exciting daughter waves in a successive way. The fluctuations can appear as sideband waves associated with the linear modes. The fundamental or the most likely process to generate the sideband waves is the three-wave coupling. The coupling itself is a coherent process such that the frequencies, the wavevectors, and the initial phases of the interacting and the generated waves must fulfill the wave resonance conditions, ω₁ + ω₂ = ω₃ for the frequencies, k→1+k→2=k→3 for the wavevectors, ϕ₁ + ϕ₂ = ϕ₃ for the initial phases. Therefore, some mechanism is needed to make the wave phase random or incoherent such as dissipation of the fluctuation energy or thermal motion of the plasma. A coupling of four waves (such as wave transmission and wave reflection out of two interacting waves) or even more participating waves is possible but the probability becomes lower when more participating waves are involved in the interactions.

The three-wave coupling is irrelevant from the physics of the linear modes. If the daughter wave happens to have a frequency of some linear modes (onto the same branch or a different branch), the daughter wave can be supported by the plasma as a background medium, and can propagate over a large distance and can exist for a longer time. Sideband waves are formed if there is a frequency mismatch between the daughter wave and the linear modes, and if there is a continuous pumping of the daughter wave. Sideband or nonlinear modes can be any propagating wave components other than the linear mode fluctuations. The lifetime of the sideband waves plays an important role in turbulence evolution. Sideband waves may break into other frequencies and wavevectors through successive wave-wave interactions.

Wave couplings can happen even for smaller fluctuation amplitudes because the nonlinear effect caused by a spatial gradient (or a wavenumber) can compete against the small amplitudes. In the fluid picture, the nonlinear terms such as the advection or the Lorentz force represent a combination of (1) self-coupling between different scales of the flow velocity or the magnetic field and (2) mediation by a wavenumber (which comes from the spatial derivative, the nabla operator).

Using four-point tetrahedral measurements of Cluster, both the frequencies and the wavevectors in the three-dimensional space can be determined in the solar wind. Furthermore, the frequencies can be corrected for the Doppler shift (a product of the wavevector with the flow velocity vector) and the frequencies can be studied in the plasma rest frame co-moving with the plasma flow. Figure 2 left exhibits a histogram of the frequency-wavevector occurrence in the co-moving frame using 9 time intervals of Cluster's solar wind measurements and 512 discrete wave components therein [44]. Frequencies are normalized to the proton cyclotron frequency, and the wavenumbers (the magnitude of the wavevector) are normalized by multiplying the proton inertial length (Alfvén speed divided by the proton cyclotron frequency). Comparison with the theoretically estimated or expected dispersion relations shows no clear or strong agreement in the wave data with the linear mode waves such as the kinetic Alfvén mode or the ion Bernstein modes. The majority of the detected waves in the solar wind do not agree with the frequencies of the linear mode, but appear either as sideband waves associated with the linear mode or as a concentration below the ion gyro-frequency that does not belong to any linear modes. The sideband formation around the linear mode is also indicated on other time intervals of the Cluster data, e.g., around the frequencies of the kinetic Alfvén mode [45–47].

The existence of sideband waves is also confirmed in the hybrid plasma simulations (Figure 2 right). Moreover, the frequency spread around the linear mode grows together with the turbulence evolution. That is, the sideband spread serves as an index of turbulence evolution for distinct linear modes [48–50]. The sideband spread can be measured by computing the variance of the fluctuation energy around the linear mode frequency as: (1)Δ=∫​ω−ω+(ω−ω0(k))2E(k,ω)dω where ω₀ denotes the linear mode frequency at a given value of the wavenumber and E(k, ω) the fluctuation energy as a function of the frequency ω and the wavenumber k. If multiple wave modes co-exist in the turbulence data, the frequency range for the integration needs to be specified such as ω₋ and ω₊ for the lower and upper limits. Hybrid plasma simulations show that the sideband activity evolves when the fluctuations evolve into turbulence by exciting waves at smaller wavelengths and that the evolution profile depends on the wave modes [48, 50].

Using 31 solar wind events and 2,328 discrete waves therein, the wavenumber-frequency spectra are fitted into a scaling of the frequency deviation (as measured by the standard deviation) as σ ∝ k^1.6 such that the frequency deviation is inflated more rapidly at higher wavenumbers [47]. A hydrodynamic scaling of the frequency deviation is σ ∝ k^2/3 (in the Lagrangian-frame), and is asymptotic at higher wavenumbers.

So far, not many unambiguous observational studies are available which indicate the frequency deviation from the linear mode waves in the solar wind. The E-over-B method (the ratio of the electric to the magnetic field fluctuations) can determine the dispersion relation experimentally, but one has to assume that there is only one wave mode at each frequency and that the fluctuations are of electromagnetic type and not of electrostatic type. The frequency deviation is contributed not only by wave-wave interactions but also by a random sweeping by large-scale flow variations and large-scale wave motions. In future studies of kinetic plasma turbulence, the origin of the frequency deviation should be more systematically studied.

Magnetohydrodynamic (or Alfvénic) turbulence develops primarily into filaments of magnetic flux along the mean magnetic field. The reason for the filament formation lies in the fundamental wave-wave coupling of the Alfvén waves. In the MHD picture, the combination of the dispersion relation for the Alfvén mode with the three-wave coupling constrains that the parallel component of the wavevector is either zero (for one incident wave) or a constant (for the other incident wave) [51]. That is, the energy cascade is strictly perpendicular in this treatment. The wavevector of the daughter wave has a larger value of the perpendicular component of the wavevector. Also, it is worth noting that the three-wave coupling (in the wavevector sense) is successfully demonstrated in a laboratory plasma experiment [52].

In the ion-kinetic picture, propagation directions are typically given as an input to solving for the dispersion relations in the linear Vlasov theory. However, incorporation of the dispersion relation into the three-wave coupling predicts that the wavevectors evolve increasingly in the perpendicular direction. For both the whistler and the kinetic Alfvén modes, three-wave coupling favors energy cascade perpendicular to the mean magnetic field [53]. Numerical simulations of plasma turbulence indicate that the perpendicular wavevector geometry is more valid: magnetohydrodynamic simulations [54], ion kinetic or hybrid simulations [48, 55, 56], gyro-kinetic simulations [57], and particle-in-cell simulations [58–60].

Two scenarios are possible as to describing how the wavevectors are organized: parallel and perpendicular wavevector geometries. The parallel wavevector geometry (also referred to as the slab geometry) reflects the picture of packets of Alfvén waves propagating parallel and anti-parallel to the mean magnetic magnetic field with different wavelengths. The perpendicular wavevector geometry (also referred to as the quasi-two-dimensional geometry) reflects the picture of turbulence evolving in the plane perpendicular to the mean magnetic field. In terms of the Alfvén mode, the dispersion relation becomes a zero-frequency mode such that coherent structures develop in the perpendicular wavevector geometry.

Spacecraft measurements in the solar wind indicate that both of the wavevector geometries are possible: the parallel wavevector geometry on large (magnetohydrodynamic) scales (or referred to as the outer scales) [61] or high-speed streams [62], and the perpendicular wavevector geometry on small scales from magnetohydrodynamic down to ion-kinetic scales [63, 64] or low-speed streams [62].

Using the Cluster spacecraft data, the magnetic energy can be determined directly in the three-dimensional ion-kinetic wavevector domain by integrating the spectra over the frequencies. Figure 3 left displays a characteristic or typical example of the wavevector anisotropy [64]. The energy distribution extends preferentially in the perpendicular direction to the mean magnetic field. In contrast, the spectral energy falls down more quickly in the parallel direction and most of the fluctuation energy is confined to small parallel wavenumbers, that is, there are large-scale structure along the mean magnetic field and many different fluctuation scales perpendicular to the mean field.

The picture of the perpendicular wavevector geometry is also confirmed by the hybrid plasma simulations [63, 64] for different values of plasma beta observed by the Cluster spacecraft (both low-beta and high-beta plasmas). The energy spectrum (for magnetic field fluctuations) extends in the perpendicular direction to the mean magnetic field and does not develop in the parallel direction. The type of the wavevector geometry can be measured by measuring the variance of the spectral extension in the parallel and perpendicular directions, and computing the ratio as follows: (2)A=tan2ψ=∑{k‖,k⊥}k⊥2E(k⊥,k‖)∑{k‖,k⊥}k‖2E(k⊥,k‖) The degree of wavevector anisotropy A changes with the value of plasma parameter beta, that is, the wavevector anisotropy becomes weaker at higher values of ion beta. A possible interpretation is that parallel and oblique propagating waves are generated more in the high-beta plasma. Yet, it is not known if the fluctuations are more compressive in the high-beta plasma. An empirical scaling law is found to characterize the wavevector anisotropy as a function of plasma beta, A ∝ β^−0.3 [63].

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.