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, $\tau$, 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 $\tau$.
