  Kinney et al [PPL \textbf{1},(1994)] have investigated plasma filamentary
structure dynamics. More recently, Wilkin et al [Phys Rev Lett
\textbf{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 ${\textbf{E}^{3}}$. In 2D, either no dynamo action spectrum is
found, in agreement with Cowling anti-dynamo theorem, or slow dynamos [PPL
\textbf{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.
