Fluctuation dynamo at finite correlation times using renewing flows

Abstract: Fluctuation dynamos are generic to turbulent astrophysical systems. The only analytical model of the fluctuation dynamo, due to Kazantsev, assumes the velocity to be delta-correlated in time. This assumption breaks down for any realistic turbulent flow. We generalize the analytic model of fluctuation dynamo to include the effects of a finite correlation time, $\tau$, using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_L$ leads to the standard Kazantsev equation in the $\tau \to 0$ limit, and extends it to the next order in $\tau$. We find that this evolution equation involves also third and fourth spatial derivatives of $M_L$, indicating that the evolution for finite $\tau$ will be non-local in general. In the perturbative case of small-$\tau$ (or small Strouhl number), it can be recast using the Landau-Lifschitz approach, to one with at most second derivatives of $M_L$. Using both a scaling solution and the WKBJ approximation, we show that the dynamo growth rate is reduced when the correlation time is finite. Interestingly, to leading order in $\tau$, we show that the magnetic power spectrum, preserves the Kazantsev form, $M(k) \propto k^{3/2}$, in the large $k$ limit, independent of $\tau$.