The propagation of cosmic rays (CRs) in turbulent interstellar magnetic fields is typically described as a spatial diffusion process. This formalism predicts only a small deviation from an isotropic CR distribution in the form of a dipole in the direction of the CR density gradient or relative background flow. We show that the existence of a global CR dipole moment necessarily generates a spectrum of higher multipole moments in the local CR distribution. These anomalous anisotropies are a direct consequence of Liouville's theorem in the presence of a local turbulent magnetic field. We show that the predictions of this model are in excellent agreement with the observed power spectrum of multi-TeV CRs.PACS numbers: 98.70. Sa, 98.35.Eg Introduction.-The arrival directions of Galactic cosmic rays (CRs) are highly isotropic. This is expected from a diffusive propagation of CRs in the interstellar medium, where the effective scattering in turbulent magnetic fields randomizes the particle momenta over time. Diffusion theory (including also convective and dissipative processes) provides an excellent description of Galactic CR fluxes and their chemical abundances, e.g. [1]. In this framework the only deviation from an isotropic CR arrival direction is in the form of a weak dipole anisotropy. The phase and strength of this dipole is expected to be a combined effect of the relative motion of the solar system with respect to the frame where CRs are isotropic [2] and the density gradient of CRs in the direction of their sources [3][4][5].Cosmic ray anisotropies up to the level of one-per-mille have been observed at various energies by the observatories Tibet AS-γ [6,7] [17]. The explanation of the strength and phase of the observed dipole anisotropy is challenging, but is qualitatively consistent with the diffusive prediction [4]. However, some of the observations also show significant multi-TeV CR excesses at smaller angular scales with unknown origin. In particular, a high statistics sample of multi-TeV CRs seen by the IceCube observatory [16] shows significant power in small-scale multipole moments with 20 as shown in Fig. 1. It has been speculated that localized CR excesses can be a combined effect of CR acceleration in nearby supernova remnants [18] and the local intergalactic magnetic field structure introducing an energy-dependent magnetic mirror leakage [19] or preferred CR transport directions [20]. Magnetic reconnections in the heliotail [21], non-isotropic particle transport in the heliosheath [22] or the heliospheric electric field structure [23] have also been considered as a source of these small-scale anisotropies. Another variant considers the effect of magnetized outflow from old supernova remnants [24]. More exotic models invoke strangelet production in molecular clouds [25] or in neutron stars [26].