Slow plasma dynamo driven by electric current helicity in non-compact Riemann surfaces of negative curvature

Boozer addressed the role of magnetic helicity in dynamos [Phys Fluids \textbf{B},(1993)]. He pointed out that the magnetic helicity conservation implies that the dynamo action is more easily attainable if the electric potential varies over the surface of the dynamo. This provided us with motivation to investigate dynamos in Riemannian curved surfaces [Phys Plasmas \textbf{14}, (2007);\textbf{15} (2008)]. Thiffeault and Boozer [Phys Plasmas (2003)] discussed the onset of dissipation in kinematic dynamos. When curvature is constant and negative, a simple simple laminar dynamo solution is obtained on the flow topology of a Poincare disk, whose Gauss curvature is $K=-1$. By considering a laminar plasma dynamo [Wang et al, Phys Plasmas (2002)] the electric current helicity ${\lambda}\approx{2.34m^{-1}}$ for a Reynolds magnetic number of $Rm\approx{210}$ and a growth rate of magnetic field $|{\gamma}|\approx{0.022}$. Negative constant curvature non-compact $\textbf{H}^{2}$, has also been used in one-component electron 2D plasma by Fantoni and Tellez (Stat Phys, (2008)). Chicone et al (CMP (1997)) showed fast dynamos can be supported in compact $\textbf{H}^{2}$. PACS: 47.65.Md. Key-word: dynamo plasma.