Toward an asymptotic behaviour of the ABC dynamo

TL;DR

This study uses high-resolution simulations to investigate the asymptotic behavior of the ABC dynamo, revealing that the growth rate still depends on the magnetic Reynolds number even at very high values, challenging previous assumptions.

Contribution

It provides the first high-resolution simulations up to Rm=5×10^5, showing the absence of an asymptotic regime and suggesting complex spectral behavior.

Findings

Growth rate decreases with increasing Rm at high values

No asymptotic regime observed up to Rm=5×10^5

Possible indication of mode crossing or essential spectrum approach

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 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.