## 1. Introduction

[2] 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.* [2009], *Svidzinski et al.* [2009], and *Dmitruk and Matthaeus* [2009], finding only weak evidence for wave behavior.

[3] At very low frequencies (smaller than the proton gyro-frequency, Ω_{p}) 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 On sufficiently small scales, the MHD picture is no longer valid, and particle kinetic effects must be taken into account.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.* [2010] 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].

[4] 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.