A generalized model for small scale dynamo at finite correlation times

Fluctuation (or small scale) dynamos are generic in astrophysical plasmas. The Kazantsev model of the fluctuation dynamo assumes a delta-correlated in time velocity field, which is unrealistic. Using renewing flows with finite time correlation, $\tau$, we derive a generalized model of the fluctuation dynamo. We recover the standard Kazantsev equation for the evolution of longitudinal magnetic correlation, $M_L$, for $\tau \to 0$. The generalized equation involves third and fourth spatial derivatives of $M_L$ indicating nonlocality. We solve the equation in the large $k$ limit and by using Landau-Lifschitz approach, we recast the equation to one which involves 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 $\tau$.