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tokens: are denoted by black vertical lines. In all cases of Figure 1 a spectral break is observed at the largest of the three scales, i.e., the scale corresponding to the lowest ion frequency . This result is consistent with the observations in the solar wind (Chen et al. 2014 ). However, in the range , where is the frequency corresponding to the smallest ion scale, the power spectra are modified by ion instabilities: while a power law seems to develop after the break in the case of very high beta (Figure 1 , top panel), the onset of the cascade is shifted toward higher frequencies (Figure 1 , middle and bottom panels) by the formation of a broader “knee” and develops at . For example, in the bottom panel, we observe a local bump at the break, which has been identified in this particular event (see Alexandrova et al. 2006 ) as the signature of Alfvén vortices, controlled by . Each case presents a different slope α in this range, as shown by the compensated spectra depicted by red lines in each panel (the spectral index α is indicated in each panel on the right). Kinetic instabilities can modify the power spectrum at ion scales and affect the fluctuations in the small-scale range. Therefore, the identification of kinetic instabilities is important and it is the subject of investigation in the following section. 3. SMALL-SCALE TURBULENCE ANISOTROPY AND SPECTRAL INDEX In this section we study in detail the effects of mirror modes and AIC waves, which are observed to dominate in the magnetosheath (Schwartz et al. 1996 ), on the spectral index and anisotropy of fluctuations in the small-scale range, i.e., f ∈ [0.2–10] Hz in our case. To do so, we choose four time intervals of about 3–5 minutes each (i.e., PSDs at frequencies 0.025 Hz are not reliable) and we sort them out depending on the nature of the present waves. The selected cases are chosen as follows: (1) no waves are dominating the spectrum (2008 March 17, 12:30–12:35 UTC); (2) only strong mirror modes are observed (2008 March 25, 07:01–07:05 UTC); (3) strong AIC waves dominate (2008 March 30, 01:35–01:39 UTC); and (4) a mixture of the mirror and AIC waves is observed (2008 March 25, 06:51–06:56 UTC). These intervals are selected during periods of low v ( v ≈ 50 km s −1 ) so that the effects of the Doppler shift due to the plasma flow past the spacecraft (i.e., increase of δB y ) are minimized (Sahraoui et al. 2006 ; Alexandrova et al. 2008b ). Because of the low bulk velocity Taylor's hypothesis could be violated (see, e.g., Klein et al. 2014 ), thus we performed three tests to ensure that this hypothesis holds in the selected cases. First, we determined the turbulence intensity ( v 0 being the mean bulk velocity and v rms the root mean square of v fluctuations) to be in the range [0.04–0.09] (i.e., I ≪ 1); then, we found that the relation is fulfilled (for a detailed description of the test, see Sahraoui et al. 2012 ); and finally, we found good agreement between the probability density