Kinematic dynamo in a tetrahedron of Fourier modes

Considering low-order systems of kinematic dynamo, we look for the lowest possible system order which can lead to a dynamo instability. For that we decompose, in Fourier space, both velocity and magnetic fields into helical modes. Starting with a single triad, which is the lowest possible order system, we show that a dynamo instability cannot occur, unless both velocity and magnetic fields are decimated. Decimation means that only one helical mode per wave number is kept, which is unlikely in a physically realizable situation. The next possible system order is the one composed of a set of four triads forming a tetrahedron. In that case we show that a dynamo instability is possible, without needing to decimate either the velocity field or the magnetic field. Finally we find that dynamo action is not possible if the kinetic helicity is zero at each wave number.