  The ABC flow was originally introduced by Arnol'd to investigate Lagrangian
chaos. It soon became the prototype example to illustrate magnetic-field
amplification via fast dynamo action, i.e. dynamo action exhibiting
magnetic-field amplification on a typical timescale independent of the
electrical resistivity of the medium. Even though this flow is the most
classical example for this important class of dynamos (with application to
large-scale astrophysical objects), it was recently pointed out (Bouya Isma\"el
and Dormy Emmanuel, Phys. Fluids, 25 (2013) 037103) that the fast dynamo nature
of this flow was unclear, as the growth rate still depended on the magnetic
Reynolds number at the largest values available so far $(\text{Rm} = 25000)$ .
Using state-of-the-art high-performance computing, we present high-resolution
simulations (up to 40963) and extend the value of $\text{Rm}$ up to $
5\cdot10^5$ . Interestingly, even at these huge values, the growth rate of the
leading eigenmode still depends on the controlling parameter and an asymptotic
regime is not reached yet. We show that the maximum growth rate is a decreasing
function of $\text{Rm}$ for the largest values of $\text{Rm}$ we could achieve
(as anticipated in the above-mentioned paper). Slowly damped oscillations might
indicate either a new mode crossing or that the system is approaching the limit
of an essential spectrum.
