Toward an asymptotic behaviour of the ABC dynamo

¹ KTH Mechanics - Osquars Backe 18, SE-100 44 Stockholm, Sweden
² UMR 7586 Institut de mathématiques de Jussieu, Analyse fonctionnelle, Université Paris Diderot 2 Avenue de France, F-75013 Paris, France
³ MAG (ENS/IPGP), LRA, Département de Physique, Ecole Normale Supérieure - 24, rue Lhomond, F-75231 Paris Cedex 05, France

Received: 16 January 2015
Accepted: 1 April 2015

Abstract

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 Ismaël 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 . Using state-of-the-art high-performance computing, we present high-resolution simulations (up to 4096³) and extend the value of up to . 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 for the largest values of 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.