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Propagation of fast particles in turbulent magnetic fields is a fascinating research field with many facets that still have to be answered. The proper description of cosmic-ray transport is of utmost importance to understand the signatures from cosmic-ray electrons and hadrons, as well as gamma rays and neutrinos. The turbulence governs the particle transport in galactic outflows and in the interstellar medium, typically described via diffusion tensors in the transport equations. The resulting cosmic-ray diffusion not only influences signatures of cosmic-ray anisotropy, but even the type and strength of the turbulence influence the energy spectra of cosmic rays.

Important questions related to the cosmic-ray diffusion in possibly intermittent, turbulent magnetic fields are:

- How does turbulence determine the transport of particles and at which energies is this transport diffusive, superdiffusive or subdiffusive?
- How does a 'phase diagram' of transport as a function of particle energies look like?
- Do strong current sheets and intermittency affect particle transport?
- In case of diffusion, what is the structure of the diffusion tensor?
- What is the impact of the large-scale structures in which the turbulence is embedded?

The most natural approach to the study of particle transport in turbulence would consist in a numerical solution of the magnetohydrodynamic (MHD) equations and the propagation of particles in the dynamical turbulent MHD fields. This approach has been applied to simulate small regions of interest to extract processes and effective diffusion coefficients as input for coarse-grained transport descriptions. The dream to describe significantly larger regions of interest (e.g., the heliosphere, see below, the interstellar medium, or galactic outflows) in this way will, however, remain one for the foreseeable future: for simulations covering, e.g., the whole heliosphere one has to resolve small scales at the ion-gyro radius (~ 10^{5} m), where damping processes operate, up to about 100 AU, which is the distance to the solar wind termination shock. This would result in a mesh of about 1.500.000^{3} points, a size which is not possible to handle in the near future. To circumvent this problem, different strategies involving the generation of synthetic turbulence fields have been designed, as described below. However, all the methods presented so far have shortcomings that may affect simulation results to different degrees. For example, most methods only require as input the shape of the energy spectrum of the synthetic turbulence (e.g. that of a Kolmogorov spectrum). However, fields with localized structures and Gaussian turbulent fields can exhibit an identical energy spectrum, while particles moving within will have quite different transport properties. Therefore, a central question is for which physical parameters (particle energy, turbulence strength relative to the background magnetic field, scales) a description of the turbulence with a Gaussian random field of given spectrum is sufficient and, conversely, where non-Gaussian (e.g. intermittent) random fields have to be considered. An excellent testbed for theories is the cosmic-ray transport in the heliosphere, i.e. the region around the Sun occupied by the turbulent solar wind plasma, which is well-observed with a fleet of spacecraft. There, decades of in situ observations have allowed both to measure turbulence properties and to model the transport of energetic particles within. In modeling synthetic turbulence, calculating anisotropic diffusion tensors and cross-checking the theoretical results in the heliosphere, we will provide the microscale properties of cosmic-ray diffusion to the cosmic-ray modeling in galactic environments in order to understand the outflow environments of galactic systems.

- Prof. Dr. R. Grauer
- PD Dr. H. Fichtner