Dynamo in the Presence of Differential Rotation

Abstract

It is well know that the ‘dynamo’ theory has a number of vetoes; e.g. axisymmetric, two-dimensional, central-symmetric, etc. dynamo are impossible. In principle, the problem is essentially three-dimensional in any coordinate system. This is the main difficulty of both the theory itself and its possible applications. In fact, one prefers to believe that, for example, a non-rigid body-rotating star or convection in the Earth's nucleus possesses axis symmetry. However, due to the above vetoes one has to add finer effects (Coriolis strength, density, inhomogeneity) to create asymmetrical convection. On the other hand, the authors try to find the most simple movements with minimum deviations from axial symmetry. Thus, the Herzenberg's dynamo (Herzenberg, 1958) is realized by two rotating cylinders, axes of which are parallel to each other (see also Galaitis, 1973; Galaitis and Freinberg, 1974), the Lortz's dynamo-spiral movement (Lortz, 1968; Ponomarenko, 1973). Nevertheless, the mentioned vetoes possess a common feature, the assumption regarding the symmetry extends both to the movement and to the field. Hence, it makes sense to raise a question whether symmetric movements are able to generate an asymmetric field. A positive answer to this question, in particular, is given by Tverskoy's model (Tverskoy, 1966) – the toroidal vortex. The latter possesses axial symmetry. Nevertheless, the toroidal vortex is a complex motion; we will proceed along the path of a minimum simplification.