Comment on paper by L. M. Malyshkin and S. Boldyrev, "Magnetic dynamo action at low magnetic Prandtl numbers", PRL 105, 215002 (2010)

TL;DR

This paper critically examines the scaling law for small-scale dynamo growth rates at low magnetic Prandtl numbers, revealing that different asymptotic behaviors occur near and far from the threshold, contrary to previous claims.

Contribution

It clarifies the asymptotic behavior of the dynamo growth rate, showing two distinct regimes near and far from the threshold, challenging prior assumptions.

Findings

Two different asymptotics for the dynamo growth rate near and far from the threshold.

The scaling mbda sim Rm^{1/2} is not valid near the threshold.

Numerical analysis supports the existence of separate regimes.

Abstract

Is the scaling, \lambda \sim Rm^{1/2}, for the growth rate of small-scale dynamo instability at low magnetic Prandtl numbers and large magnetic Reynolds numbers, Rm, valid in the vicinity of the threshold? Our analysis and even numerical solution (Malyshkin and Boldyrev, 2010) of the dynamo equations for a Gaussian white-noise velocity field (the Kazantsev-Kraichnan model) imply that the answer is negative. Contrary to the claim by Malyshkin and Boldyrev (2010), there are two different asymptotics for the dynamo growth rate: in the vicinity of the threshold and far from the threshold.