Recent results concerning the amplification of magnetic field frozen to a two-dimensional spatially periodic flow consisting of two distinct pulsed Beltrami waves are summarised. The period a of each pulse is long (α ≫ 1) so that fluid particles make excursions large compared to 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 (Z) produced. Under the assumption, (Tk+2Z) − |λ∈|2«TkZ) → 0 as K → ∈, an asymptotic representation of the complex constant λ∈ is obtained, which determines the growth rate α−1(α|λ∈|). The main result is the construction of a family of smooth vector fields ZN and complex constants λN with the properties (for even N), and for all integers K(> 0), where ∈ = α−3/2. The relation of ZN and λN to the modes of the corresponding dissipative problem with the fastest growth rates is discussed.