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Effects of finite time correlation on the Kazantsev model of the fluctuation dynamo
The only analytical model of fluctuation dynamo
is the Kazantsev model which assumes delta correlated in time velocity field.
Here we generalise the Kazantsev model to finite correlation time, τ,
by modelling the velocity field as a renovating flow. We recover the Kazantsev
equation for the longitudinal magnetic correlation function, ML in the limit of τ→0.
On extending this equation to the next order in τ
to include the effects of finite correlation in time, we obtain third
and fourth derivatives of ML as perturbative terms in τ.
Using the Landau-Lifschitz approach the evalution
equation for ML, can be recast into one which involves at most second derivatives of ML.
From an asymptotic treatment of the resulting equation,
we show that the magnetic power spectrum, M(k), remains exactly(!) the
Kazantsev spectrum of M(k) ∝ k3/2, even at finite correlation times.