Revisiting the ABC flow dynamo

The Arnol'd-Beltrami-Childress flow is a prototype for fast dynamo action, essential to the origin of magnetic field in large astrophysical objects. Probably the most studied configuration is the classical 1 : 1 : 1 flow. We investigate its dynamo properties varying the magnetic Reynolds number Rm. We identify two kinks in the growth rate, which correspond, respectively, to an eigenvalue crossing and to an eigenvalue coalescence. The dominant eigenvalue becomes purely real for a finite value of the control parameter. Finally, we show that even for Rm = 25 000, the dominant eigenvalue has not yet reached an asymptotic behaviour. It still varies very significantly with the controlling parameter. Even at these very large values of Rm the fast dynamo property of this flow cannot yet be established.