Understanding helical magnetic dynamo spectra with a non-linear four-scale theory

Recent magnetohydrodynamic (MHD) dynamo simulations for magnetic Prandtl number > 1 demonstrate that when MHD turbulence is forced with sufficient kinetic helicity, the saturated magnetic energy spectrum evolves from having a single peak below the forcing scale to become doubly peaked with one peak at the system (= largest) scale and one at the forcing scale. The system scale field growth is well modelled by a recent non-linear two-scale helical dynamo theory in which the system and forcing scales carry magnetic helicity of opposite sign. However, a two-scale theory cannot model the shift of the small-scale peak toward the forcing scale. Here I develop a four-scale helical dynamo theory which shows that the small-scale helical magnetic energy first saturates at very small scales, but then successively saturates at larger values at larger scales, eventually becoming dominated by the forcing scale. The transfer of the small-scale peak to the forcing scale is completed by the end of the kinematic growth regime of the large-scale field, and does not depend on magnetic Reynolds number R-M for large R-M. The four- and two-scale theories subsequently evolve almost identically, and both show significant field growth on the system and forcing scales that is independent of R-M. In the present approach, the helical and non-helical parts of the spectrum are largely decoupled. Implications for fractionally helical turbulence are discussed.