Saturation of turbulent helical dynamos

The presence of large scale magnetic fields in nature is often attributed to the inverse cascade of magnetic helicity driven by turbulent helical dynamos. In this work we show that in turbulent helical dynamos, the inverse flux of magnetic helicity towards the large scales $\Pi_{\mathcal{H}}$ is bounded by $|\Pi_{\mathcal{H}}|\le c \epsilon k_\eta^{-1}$, where $\epsilon$ is the energy injection rate, $k_\eta$ is the Kolmogorov magnetic dissipation wavenumber and $c$ an order one constant. Assuming the classical isotropic turbulence scaling, the inverse flux of magnetic helicity $\Pi_{\mathcal{H}}$ decreases at least as a $-3/4$ power-law with the magnetic Reynolds number $Rm$ : $|\Pi_{\mathcal{H}} | \le c \epsilon \ell_f Rm^{-3/4}\max[Pm,1]^{1/4}$, where $Pm$ the magnetic Prandtl number and $\ell_f$ the forcing lengthscale. We demonstrate this scaling with $Rm$ using direct numerical simulations of turbulent dynamos forced at intermediate scales. The results further indicate that nonlinear saturation is achieved by a balance between the inverse cascade and dissipation at domain size scales $L$ for which the saturation value of the magnetic energy is bounded by ${\mathcal{E}}_\text{m}\leq c L (\epsilon \ell_f)^{2/3} Rm^{1/4}\max[1,Pm]^{1/4}$. Numerical simulations also demonstrate this bound.