Fluctuation dynamo at finite correlation times and the Kazantsev spectrum

TL;DR

This paper extends the Kazantsev model of fluctuation dynamo to include finite correlation times of the velocity field, showing that the magnetic power spectrum remains unchanged in the large wavenumber limit.

Contribution

It derives a generalized fluctuation dynamo model with finite correlation time using renovating flows, recovering the Kazantsev equation in the limit and analyzing the spectrum's robustness.

Findings

The generalized model includes higher-order derivatives in $ au$.

The magnetic power spectrum remains $k^{3/2}$ at large $k$, independent of $ au$.

The model reduces to the classic Kazantsev equation as $ au o 0$.

Abstract

Fluctuation dynamos are generic to astrophysical systems. The only analytical model of the fluctuation dynamo is Kazantsev model which assumes a delta-correlated in time velocity field. We derive a generalized model of fluctuation dynamo with finite correlation time, $\tau$, using renovating flows. For $\tau \to 0$, we recover the standard Kazantsev equation for the evolution of longitudinal magnetic correlation, $M_L$. To the next order in $\tau$, the generalized equation involves third and fourth spatial derivatives of $M_L$. It can be recast using the Landau-Lifschitz approach, to one with at most second derivatives of $M_L$. Remarkably, we then find that the magnetic power spectrum, remains the Kazantsev spectrum of $M(k) \propto k^{3/2}$, in the large $k$ limit, independent of $\tau$.