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, $\tau$, 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 $\tau$.
