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Kinney et al [PPL \textbf{1},(1994)] have investigated plasma filamentarystructure dynamics. More recently, Wilkin et al [Phys Rev Lett\textbf{99}:134501,(2007)] have shown that kinetic energy spectrum of magneticstructures in small-scale dynamos, are predominantly filamentary. Kirilov et al[PRE (2009)] have shown that use of the boundary values of the mean-fieldisotropic helical turbulent ${\alpha}^{2}$-dynamo, could determine themorphology of dynamo instability. In this paper, filamentary Frenet frame indiffusive media, displays the existence of kinematic chaotic dynamo in 3DEuclidean space ${\textbf{E}^{3}}$. In 2D, either no dynamo action spectrum isfound, in agreement with Cowling anti-dynamo theorem, or slow dynamos [PPL\textbf{15},(2008)]. Curvature and diffusion effects are encodded in the matrixrepresentation of ${\alpha}^{2}$-dynamo operator. Instead of principal scalarcurvatures ${\kappa}_{1}$ and ${\kappa}_{2}$ of the surface of structures, onlyone scalar curvature ${\kappa}$ is needed to determine dynamos spectra.Filament thickness, increases with scalar curvature, as happens in solarphysics.

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