Memory effects in turbulent transport

TL;DR

This paper investigates how turbulent transport in magnetic fields and passive scalars exhibits memory effects, depending on the past history of the mean field, and introduces integral kernels to model these effects.

Contribution

The study demonstrates the presence of memory effects in turbulent transport and develops analytical models for integral kernels to better predict mean field growth or decay.

Findings

Memory effects depend on the frequency or growth rate of the mean field.

Integral kernels can accurately describe the characteristic timescale of memory effects.

Steady state transport coefficients may misestimate dynamo growth rates.

Abstract

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.