  In the mean-field theory of magnetic fields, turbulent transport, i.e. the
turbulent electromotive force, is described by a combination of the alpha
effect and turbulent magnetic diffusion, which are usually assumed to be
proportional respectively to the mean field and its spatial derivatives. For a
passive scalar there is just turbulent diffusion, where the mean flux of
concentration depends on the gradient of the mean concentration. However, these
proportionalities are approximations that are valid only if the mean field or
the mean concentration vary slowly in time. Examples are presented where
turbulent transport possesses memory, i.e. where it depends crucially on the
past history of the mean field. Such effects are captured by replacing
turbulent transport coefficients with time integral kernels, resulting in
transport coefficients that depend effectively on the frequency or the growth
rate of the mean field itself. In this paper we perform numerical experiments
to find the characteristic timescale (or memory length) of this effect as well
as simple analytical models of the integral kernels in the case of passive
scalar concentrations and kinematic dynamos. The integral kernels can then be
used to find self-consistent growth or decay rates of the mean fields. In
mean-field dynamos the growth rates and cycle periods based on steady state
values of alpha effect and turbulent diffusivity can be quite different from
the actual values.
