An explicit solution for static unbounded helical dynamos

The Lortz dynamo with helical symmetry is re-examined. It is shown that by imposing appropriate boundary conditions the set of possible solutions can be broken down into various classes characterized by the behavior of the mean magnetic field. It is found that, as the cylindrical radius, s, tends to zero, ∼ 0(s j ), ∼ const + 0(s j−i ), where j>5. It is proved that the azimuthal wavenumber associated with the j=5 class is necessarily equal to 2. The existence of at least one cylindrical surface inside which the dynamo is self-sustained is demonstrated. A new simple explicit solution is obtained. The topology the magnetic field is studied and three-dimensional pictures of the magnetic field lines are exhibited. Finally, a criterion for reversal of the magnetic field as a function of radius is ohtained and is applied to our solution.