Magnetic filamentary structures in the ${\alpha }^2$ dynamo spectrum in plasmas

Kinney et al [PPL $\mathbf{\text{1}}$,(1994)] have investigated plasma filamentary structure dynamics. More recently, Wilkin et al [Phys Rev Lett $\mathbf{\text{99}}$:134501,(2007)] have shown that kinetic energy spectrum of magnetic structures in small-scale dynamos, are predominantly filamentary. Kirilov et al [PRE (2009)] have shown that use of the boundary values of the mean-field isotropic helical turbulent ${\alpha }^2$-dynamo, could determine the morphology of dynamo instability. In this paper, filamentary Frenet frame in diffusive media, displays the existence of kinematic chaotic dynamo in 3D Euclidean space ${\mathbf{\text{E}}}^3$. In 2D, either no dynamo action spectrum is found, in agreement with Cowling anti-dynamo theorem, or slow dynamos [PPL $\mathbf{\text{15}}$,(2008)]. Curvature and diffusion effects are encodded in the matrix representation of ${\alpha }^2$-dynamo operator. Instead of principal scalar curvatures ${\kappa }_1$ and ${\kappa }_2$ of the surface of structures, only one scalar curvature $\kappa$ is needed to determine dynamos spectra. Filament thickness, increases with scalar curvature, as happens in solar physics.