Observations of young supernova remnants (SNRs) in X-rays and gamma-rays have provided conclusive evidence for particle acceleration to at least TeV energies. Analysis of high spatial resolution X-ray maps of young SNRs has indicated that the particle acceleration process is accompanied by strong non-adiabatic amplification of magnetic fields. If Fermi acceleration is the mechanism producing the energetic cosmic rays (CRs), the amplified magnetic field must be turbulent and CR-driven instabilities are among the most probable mechanisms for converting the shock ram pressure into the magnetic turbulence. The development and evolution of strong magnetic turbulence in the collisionless plasmas forming SNR shells are complicated phenomena which include the amplification of magnetic modes, anisotropic mode transformations at shocks, as well as the nonlinear physics of turbulent cascades. Polarized X-ray synchrotron radiation from ultra-relativistic electrons accelerated in the SNR shock is produced in a thin layer immediately behind the shock and is not subject to the Faraday depolarization effect. These factors open possibilities to study some properties of magnetic turbulence and here we present polarized X-ray synchrotron maps of SNR shells assuming different models of magnetic turbulence cascades. It is shown that different models of the anisotropic turbulence can be distinguished by measuring the predominant polarization angle direction. We discuss the detection of these features in Tychos SNR with the coming generation of X-ray polarimeters such as the Imaging X-ray Polarimetry Explorer (IXPE).
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