Frequency versus wave number diagram of turbulent magnetic fluctuations in the solar wind was determined for the first time in the wide range over three decades using four Cluster spacecraft. Almost all of the identified waves propagate quasi-perpendicular to the mean magnetic field at various phase speeds, accompanied by a transition from the dominance of outward propagation from the Sun at longer wavelengths into mixture of counter-propagation at shorter wavelengths. Frequency-wave number diagram exhibits largely scattered populations with only weak agreement with magnetosonic and whistler waves. Clear identification of a specific normal mode is difficult, suggesting that nonlinear energy cascade is operating even on small-scale fluctuations.
 The solar wind is believed to represent a fully-developed turbulent plasma for the reason that fluctuations of the interplanetary magnetic field often exhibit a power-law energy spectrum with the index close to −5/3, reminiscent of the inertial-range spectrum of Kolmogorov's theory for hydrodynamic turbulence (see recent review by Petrosyan et al. [2010, and references therein]). Fluctuations of the interplanetary magnetic field do not always reach the level of the local mean field, and the concept of dispersion relation and associated normal modes, naïvely speaking, appears to be valid for sufficiently small amplitudes. In a turbulent medium, however, separation between the mean field and the fluctuating field is scale-dependent and nonlinearities can remain even for small amplitudes. For these reasons we ask: “Is there any evidence for a dispersion relation in solar wind turbulence?” Plasma turbulence is a wide-spread phenomenon in astrophysical systems, e.g., accretion disks, interstellar medium, stellar convection zones, so that our question is essential for understanding the nature of this phenomenon. This question has been addressed in various numerical simulations, e.g., Parashar et al. , Svidzinski et al. , and Dmitruk and Matthaeus , finding only weak evidence for wave behavior.
 At very low frequencies (smaller than the proton gyro-frequency, Ωp) and on large spatial scales (larger than the proton gyro-radius or inertial length), plasma turbulence should be described by magnetohydrodynamics (MHD), the fluid picture of plasma. Linear MHD theory predicts the existence of three normal modes: the incompressible shear Alfvén mode, the compressible ‘slow’ or ion acoustic mode, and the fast magnetosonic mode which exhibits both Alfvénic and acoustic properties. On sufficiently small scales, the MHD picture is no longer valid, and particle kinetic effects must be taken into account. Recent studies of solar wind turbulence showed that the magnetic field fluctuations exhibit two distinct spectral breaks and power-laws around 0.1–1 and 10–100 Hz in the spacecraft frame of reference [Sahraoui et al., 2009], while Perri et al.  argue the spectral break around 0.1 Hz seems to be independent from proton gyro-frequency or -radius. The relative amplitude of electric to magnetic field fluctuation increases at higher frequencies [Bale et al., 2005], suggesting turbulence becomes more electrostatic in nature, but the existence of dispersion relation at high frequencies is still questionable [Matthaeus et al., 2008].
 In this paper we investigate the possible relations between frequency and wave number in solar wind turbulence for the first time in the wide range over three decades of frequencies and wave numbers. The Cluster mission [Escoubet et al., 2001] is suitable for such a task, since its four-point measurements allow us to determine dispersion relations in three-dimensional space experimentally. We use the high-resolution wave-vector analysis method called the MSR technique (Multi-point Signal Resonator) [Narita et al., 2010], and look for dispersion relations in solar wind turbulence at three distinct spatial scales using Cluster: 10,000, 1000, and 100 km.
2. Event Selection and Data Analysis
 The three selected time intervals of magnetic field data of Cluster FGM [Balogh et al., 2001] in the solar wind are: (interval 1) January 16, 2006, 0430–0630 UT during the mission phase with about 10,000 km spacecraft separation; (2) March 21, 2005, 1800–2000 UT with about 1000 km separation; and (3) February 20, 2002, 1930–2000 UT with about 100 km separation. These intervals were selected using the following criteria: (a) Cluster forms an almost regular tetrahedron thereby minimizing the spatial aliasing effects in the analysis [Narita and Glassmeier, 2009; Sahraoui et al., 2010a]; (b) the intervals are uncontaminated by ions backstreaming from the bow shock and contain very few discontinuities; (c) the local mean fields (magnetic field, flow velocity, and density) can be established as nearly constant. (d) conditions of the flow speed, the magnetic field magnitude, and the plasma parameter beta are similar as shown in Table 1. Figure 1 displays the time series plots of the magnetic field magnitude in the selected intervals. Frequency spectra of magnetic field fluctuations exhibit power-law behavior with the index close to −5/3 at frequencies up to almost 1 Hz in all three intervals, confirming the typical frequency spectra in the solar wind.
Table 1. Mean Values of Plasma and Magnetic Field Data in the Analyzed Time Intervals: Characteristic Tetrahedron Size of Cluster L, Flow Speed V, Ion Number Density n, Magnitude of Magnetic Field B, Ion Temperature T, Plasma Parameter Beta (for Ions)
 The data analysis makes extensive use of the MSR technique. This was developed as a further refinement of the wave telescope (or k-filtering, hereafter WT) technique [Glassmeier et al., 2001, and references therein]. Both techniques are an estimator of wave energy as a function of frequency and wave vector for Cluster magnetometer data, and make extensive use of the 12-by-12 cross spectral density (CSD) matrix (4 spacecraft and 3 field components) on the assumption of the plane wave geometry. MSR is different from WT in that it uses the MUSIC algorithm (MUltiple SIgnal Classification) developed by Schmidt  as further filtering into the wave vector domain to provide higher-resolution in wave vector by analyzing eigenvalues and eigenvectors of the CSD matrix and classifying them into signal and noise parts. MSR is subject to the same limitations as WT including the wave number range and spatial aliasing.
Figure 2 displays a comparison of the two spectra determined for the magnetic field data in Interval 2 at frequency 23.2 mHz. The spectrum derived by WT (upper curve) is rather flat and exhibits a peak (local maximum in the energy distribution) at a wave number of about 9.5 × 10−4 rad/km, while the one derived by MSR (lower curve) clearly identifies two distinct peaks, one at the same wave number and the other at about 3.0 × 10−4 rad/km. This second peak appears in the WT spectrum as only a spectral break. The difference in the two spectral curves represents the isotropic fluctuation in all 12 measured field components that cannot be interpreted as a plane wave. In WT all the eigenvalues of the CSD matrix are used in energy estimation, while in MSR/MUSIC only significant eigenvalues and eigenvectors are used on the assumption that small eigenvalues represent isotropic, random-phase noise. The number of the wave vectors associated with local peaks in energy distribution varies typically between one and three in MSR at a frequency of interest.
 The analysis of dispersion relation does not assume any relationship between frequency and wave vector but assumes that the local mean field is constant. The analysis procedure using MSR consists of two steps. First, we identify wave vectors that are associated with energy peaks at various frequencies in the ranges [0.002, 0.02] Hz, [0.01, 0.1] Hz, and [0.1, 1.0] Hz in the spacecraft frame of reference in Interval 1, 2, and 3, respectively. Second, the set of frequencies and wave vectors are transformed into the plasma rest frame by correcting for the Doppler shift. Ion bulk velocity obtained by the ion spectrometer on board Cluster [Rème et al., 2001] was used for Doppler correction. Our dispersion analysis is essentially the same as that used by Sahraoui et al. [2010b] except for the spectral estimator.
 Three major results are obtained from the analysis: (1) nearly-perpendicular propagation on various spatial scales; (2) no clear dispersion relation accompanied by a weak signature of magnetosonic/whistler waves; and (3) transition from anti-sunward propagation at longer wavelengths to counter-propagation (both sunward and anti-sunward) at shorter wavelengths. These properties are discussed in more detail below.
Figure 3 (top) displays the distribution of propagation angles in the wave number domain normalized using the proton inertial length. The data from the three intervals are put together with the asterisk, plus, and diamond symbols for Cluster separations at 10,000, 1000, and 100 km, respectively. Figure 3 (top) exhibits that wave vectors are quasi-perpendicular to the local mean magnetic field on these scales. The averaged angle over all identified waves is 87.7 degrees from the local mean field. Scattering or spread from the perpendicular direction is relatively small (standard deviation about 12.0 degrees) except for some deviations at higher wave numbers.
Figure 3 (bottom) displays the distribution of frequencies and wave numbers in the plasma rest frame. None of the three intervals exhibits a clear organization of dispersion relation; frequencies and wave numbers appear to be scattered. The spread in the frequency-wave number diagram can be interpreted as the sign that nonlinear energy cascade is operating. Interestingly, the distribution exhibits nevertheless a tendency that lower and higher frequencies are associated with smaller and larger wave numbers, respectively. Some waves are characterized by negative frequencies (discussed later), suggesting that solar wind turbulence contains both anti-sunward (radially outward from the Sun) and sunward (inward) propagating fluctuations in the plasma rest frame. Although the identified waves are spread in the frequency-wave number diagram, there is a weaker result concerning their dense population. Three dispersion relations are over-plotted onto the diagram using the averaged plasma beta and propagation angles from the measurements: kinetic Alfvén waves (dotted line), whistler waves (solid line), and magnetosonic waves (dashed line). Comparison yields a weak agreement with the whistler and the magnetosonic waves, in the sense that the dispersion relation overlaps with the dense parts of wave populations in the frequency-wave number diagram, and rules out the kinetic Alfvén waves in contrast to the observations of Bale et al.  and Sahraoui et al. [2009, 2010b].
 Fluctuations exhibit both positive and negative frequencies after Doppler shift correction, which means that they are propagating both anti-sunward and sunward in the plasma rest frame. The fluctuation energy differences between these two directions are displayed in Figure 4 as a function of wave numbers. The fluctuation energies are determined for the anti-sunward and the sunward propagating components (denoted by P+ and P−, respectively) in the binned wave number domain, and then their differences are computed and normalized to unity such that the value of 1 (or −1) represents propagation entirely in the anti-sunward (or sunward) direction. The 95 % confidence interval is also displayed, too, based on the average number of identified waves used for grouping and binning (about 20). Although the confidence interval is relatively large, the distribution of the energy difference between anti-sunward and sunward propagation displays two distinct populations. One is predominantly anti-sunward propagation (at or around the value of 1) and the other is propagation in both directions with slightly more power in the sunward direction. The former population dominates longer wavelengths and the latter shorter wavelengths, thus the energy difference exhibits a transition from anti-sunward propagating fluctuations at longer wavelengths into counter-propagating fluctuations at shorter wavelengths. The latter population may suggest that solar wind turbulence is evolving to more nearly directionally-balanced turbulence.
4. Discussion and Conclusions
 Identification of a clear dispersion relation is difficult in spite of small fluctuation amplitudes and quasi-perpendicular propagation to the local mean magnetic field. Furthermore, the reversal in frequency sign at shorter wavelengths indicates that fluctuations are actively generated by nonlinear energy cascade. Even small-scale fluctuations cannot be explained by a single mode, which justifies earlier turbulence studies such as reduced MHD and recent numerical simulations by Parashar et al. , Svidzinski et al. , and Dmitruk and Matthaeus . This observation is also supported by the recent study by Carbone et al.  that both right- and left-hand polarizations survive the spectral break. The study by Servidio et al.  also suggests that dispersion relation does not play any role in Hall-MHD.
 Nevertheless the dense parts of the frequency and wave number population, although only weakly, agree with the dispersion relations for magnetosonic and whistler waves. Whether the spread in frequency-wave number diagram evolved from or through the dispersion relation for these modes or it evolved without any dispersion relation cannot be answered at this stage, but our results raise more questions, in particular: under what conditions the concept of normal mode is valid, how long the lifetime of normal and non-normal modes is, and how the frequency-wave number diagram changes as fluctuations evolve into a more or fully-developed turbulent state, e.g., as a function of relative amplitudes to the local mean field.
 Because of the assumption of plane wave geometry used in the analysis, one cannot rule out the possibility of other fluctuation geometries such as eddies. Turbulent fluctuations in the solar wind may also have various origins, including coronal magnetic field structures and solar wind expansion effect. How much energy is transported by fluctuations outward and inward with respect to the sun on the coronal level would be a valuable material for further discussion.
 This work was financially supported by the Bundesministerium für Wirtschaft und Technologie and the Deutsches Zentrum für Luft- und Raumfahrt, Germany, under contract 50 OC 0901.