Non-local memory effects of the electromotive force by fluid motion with helicity and two-dimensional periodicity

In mean-field dynamo theory, the electromotive force term 〈u′× B′〉 due to small-scale fields connects the small-scale magnetic field with the large-scale field. This term is usually approximated as the α-effect, assumed to be instantaneous in time and local in space. However, the approximation is valid only when the magnetic Reynolds number Rm is much less than unity, and is inappropriate when Rm ≳1, which is the condition satisfied in the Earth's core or solar convection zone. We introduce a function φ_qr as a non-local and non-instantaneous generalization of the usual α-effect and examine its behaviour as a function of Rm in the range 1/64 ≤Rm ≤10 for a kinematic dynamo model. We use the flow of G.O. Roberts 1972 (Phil, Trans. Roy. Soc. London Ser. A, 1972, 271, 411-454), which is steady and has non-zero helicities and two-dimensional periodicity. As a result, we identify three regions in Rm space according to the behaviour of the function φ_qr: (i) Rm ≲ 1/4, where the function φ_qr is local and instantaneous and can be approximated by the traditional α and β effects, (ii) 1/4 ≲ Rm ≲ 4, where the deviation from the traditional α and β effects increases and non-localness and non-instantaneousness increase, and (iii) Rm ≳ 4, where boundary layers develop fully and non-localness and non-instantaneousness are prominent. We show that the non-local memory effect for Rm ≳ 4 strongly affects the dynamo action and explains an observed augmentation of the growth rate in the dispersion relation. The results imply that the non-local memory effect of the electromotive force should be important in the geodynamo or the solar dynamo.