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 Letter, we show that in turbulent helical dynamos, the inverse flux of magnetic helicity toward the large scales Π_{H} is bounded by |Π_{H}|≤cεk_{η}^{-1}, where ε is the energy injection rate, k_{η} is the Kolmogorov magnetic dissipation wave number, and c an order one constant. Assuming the classical isotropic turbulence scaling, the inverse flux of magnetic helicity Π_{H} decreases at least as a -3/4 power law with the magnetic Reynolds number Rm: |Π_{H}|≤cεℓ_{f}Rm^{-3/4}max[Pm,1]^{1/4}, where Pm is the magnetic Prandtl number and ℓ_{f} the forcing length scale. 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 E_{m}≤cL(εℓ_{f})^{2/3}Rm^{1/4}max[1,Pm]^{1/4}. Numerical simulations also demonstrate this bound. These results are independent of the dynamo mechanism considered. In our setup, they imply that inviscid mechanisms cannot explain large scale magnetic fields and have critical implications for the modeling of astrophysical dynamos.