Kinematic dynamo in a tetrahedron composed of helical Fourier modes

It is generally believed that helicity can play a significant role in turbulent systems, e.g. supporting the generation of large-scale magnetic fields, but its exact contribution is not clearly understood. For example there are well-known examples of large scale dynamos produced by a flow which is pointwise non-helical. In any case a break of mirror symmetry seems to be always at the heart of the dynamo mechanism. A fruitful framework to analyze such processes is the use of helical mode decomposition. In pure hydrodynamics such framework has proved its availability in study of the processes responsible for helicity cascades. It has also been used in the analysis of MHD helical mode interactions. The present work deals with the kinematic dynamo problem, solving the induction equation within the framework of helical Fourier modes decomposition. We show that the simplest modes configuration leading to an unstable solution has the form of a tetrahedron. Then the dynamo is produced by only two scales flow. We find necessary conditions for such dynamo action, not certainly related to flow helicity. The results help to understand generic dynamo flows like the one studied by G.O. Roberts (1972).