The Self - Similar Turbulent Dynamo

We consider the problem of magnetic-energy amplification by a turbulent flow of conducting fluid. This effect is known as the small-scale turbulent dynamo, and it is believed to be relevant in application to magnetic fields in astrophysical objects. It is, in fact, a generic property of random (in time and/or space) flows that they can amplify magnetic fluctuations at scales smaller than the scale of the flow itself. The amplification is a net result of stretching of the field lines by the ambient random shear associated with the flow [1, 13, 3]. The fields that are generated by this mechanism have a characteristic structure: they concentrate in flux folds containing antiparallel field lines that reverse direction at the resistive scale and remain straight up to the flow scale [7, 10, 11]. Because of the direction reversals, the magnetic energy in the wave-number space concentrates at the resistive scale [5, 6]. The properties of the small-scale dynamo are most pronounced for systems where the fluid viscosity is much larger than magnetic diffusivity (magnetic Prandtl number Prm = ν/η ≫ 1), i.e., where magnetic fields reverse direction at scales much smaller than the viscous cutoff of the fluid turbulence. This regime is realised in many astrophysical plasmas: examples are warm interstellar medium, intracluster and intergalactic plasmas, solar corona. In this note, we study the volume-filling properties of the dynamo-generated fields, i.e., the distribution of the field strength. The problem can be modelled by the equations of incompressible MHD: