MHD Turbulence, Turbulent Dynamo and Applications
  MHD Turbulence is common in many space physics and astrophysics environments.
We first discuss the properties of incompressible MHD turbulence. A
well-conductive fluid amplifies initial magnetic fields in a process called
small-scale dynamo. Below equipartition scale for kinetic and magnetic energies
the spectrum is steep (Kolmogorov -5/3) and is represented by critically
balanced strong MHD turbulence. In this paper we report the basic reasoning
behind universal nonlinear small-scale dynamo and the inertial range of MHD
turbulence. We measured the efficiency of the small-scale dynamo $C_E=0.05$,
Kolmogorov constant $C_K=4.2$ and anisotropy constant $C_A=0.63$ for MHD
turbulence in high-resolution direct numerical simulations. We also discuss
so-called imbalanced or cross-helical MHD turbulence which is relevant for in
many objects, most prominently in the solar wind. We show that properties of
incompressible MHD turbulence are similar to the properties of Alfv\'enic part
of MHD cascade in compressible turbulence. The other parts of the cascade
evolve according to their own dynamics. The slow modes are being cascaded by
Alfv\'enic modes, while fast modes create an independent cascade. We show that
different ways of decomposing compressible MHD turbulence into Alfv\'en, slow
and fast modes provide consistent results and are useful in understanding not
only turbulent cascade, but its interaction with fast particles.
