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Fluctuation dynamo at finite correlation times and the Kazantsev spectrum

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, $τ$, using renovating flows. For $τ\to 0$, we recover the standard Kazantsev equation for the evolution of longitudinal magnetic correlation, $M_L$. To the next order in $τ$, 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 $τ$.