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Fluctuation dynamo at finite correlation times using renewing flows

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, $τ$, using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_L$ leads to the standard Kazantsev equation in the $τ\to 0$ limit, and extends it to the next order in $τ$. We find that this evolution equation involves also third and fourth spatial derivatives of $M_L$, indicating that the evolution for finite $τ$ will be non-local in general. In the perturbative case of small-$τ$ (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 $τ$, we show that the magnetic power spectrum, preserves the Kazantsev form, $M(k) \propto k^{3/2}$, in the large $k$ limit, independent of $τ$.