Bounds on inertial transfer in saturated small-scale dynamos

The dimensionless dissipation coefficient $\beta_u = \varepsilon_u L_f/U^3$ is an important quantity in non-conducting turbulent flows as it relates the viscous dissipation rate $\varepsilon_u$ to the scale $L_f$ at which kinetic energy is injected into the flow and the root-mean-square velocity $U$. As $\beta_u$ expresses a relation between small-scale and large-scale dynamics, it can be interpreted as a measure of the inertial flux across scales in a turbulent flow. Here we investigate the same quantity for a saturated small-scale dynamo in order to assess the influence of a fluctuating magnetic field on the interscale inertial transfer. We obtain upper bounds on $\beta_u$ for a saturated nonhelical dynamo as a function of Reynolds and magnetic Prandtl numbers.