An asymptotic solution of a fast dynamo in a two-dimensional pulsed flow

Abstract

Abstract The amplification of magnetic field frozen to a two-dimensional spatially periodic flow consisting of two distinct pulsed Beltrami waves is investigated. The period α of each pulse is long (α » 1) so that fluid particles make excursions large compared with the periodicity length. The action of the flow is reduced to a map T of a complex vector field Z measuring the magnetic field at the end of each pulse. Attention is focused on the mean field produced. Under the assumption, −|λ7infin;|2 →0asK→∞ an asymptotic representation of the complex constant λ∞ is obtained, which determines the growth rate α−1 In (α|λ|). The main result is the construction of a family of smooth vector fields Z N and complex constants λ N which, for even N, have the properties =O(α−3 (N+2)/4 and λN-∞ = O(for all integers K (> 0). In the case of the dissipative problem at large but finite magnetic Reynolds number R(> α), it is argued that the fastest growing mode Z with amplification factor λ is appro...