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