Dynamos in cylindrical geometry

The dynamo effect is a mechanism by which mechanical energy is transformed into magnetic energy. The question as to when and how a dynamo works naturally divides into subproblems. The kinematic dynamo problem inquires whether a flowing liquid conductor amplifies a small initial magnetic seed field. The kinematic dynamo problem is in essence a stability problem, in which the magnetic field may decay to zero (stable state) or grow exponentially without limits (unstable state). The growth of the magnetic field is limited in the full dynamo problem because of the retroaction of the magnetic field on the flow field.

The classification of velocity fields into dynamos and non-dynamos is still a largely empirical matter. In most cases of interest, the stability problem has to be solved numerically. In that respect, cylindrical dynamos offer the distinctive advantage that the induction equation (a PDE) can be reduced to a small set of ordinary differential equations. The velocity fields in question are axisymmetric and consist in a translation along the axis of symmetry and a solid body rotation about that axis. The translation velocity and the angular velocity may vary radially. In the simplest case these quantities are constant within cylindrical shells but jump at the boundaries of each shell (see sketch below). These very simple models have been used to probe the minimal requirements for a flow to act as a dynamo and in order to investigate the influence of stagnant exterior conductor. Models with a radially variable conductivity have also been studied (see the references).