New results for the Herzenberg dynamo: steady and oscillatory solutions

The Herzenberg dynamo, consisting of two rotating electrically conducting spheres with non-parallel spin axes, immersed in a nite spherical conducting medium, is sim- ulated numerically for a variety of parameters not accessible to the original asymp- totic theory. Our model places the spheres in a spatially periodic box. The largest growth rate is obtained when the angle, ', between the spin axes is somewhat larger than 125. In agreement with the asymptotic analysis, it is found that the critical dynamo number is approximately proportional to the cube of the ratio of the com- mon radius of the spheres and their separation. The asymptotic prediction, strictly valid only in the limit of small spheres, remains approximately valid even when the diameter of the spheres becomes comparable to their separation. Forj'j < 90 we also nd oscillatory solutions, which were not predicted by Herzenberg's analysis. To understand such solutions we present a modied asymptotic analysis in which the separation of the two spheres is essentially replaced by the skin depth which, in turn, depends on the diameter of the spheres. The magnetic eld consists of magnetic flux rings wrapped around the two spheres. Applications to local models of turbulent dynamos and to dynamo action in binary stars are discussed.