On Large Magnetic Reynolds Number Dynamos

Braginskiiĭ's dynamo theory of the Earth's magnetic field is reviewed, with particular attention to the relative magnitudes of the symmetric and asymmetric components of the fluid flow. A general approach to the problem of the ordering of the velocity makes it clear that Braginskiĭ's choice is the only one consistent with the physical processes dominant in the Earth's core. A special class of flows, which resemble the Bullard-Gellman dynamo, is studied using Braginskiĭ's approach. By making use of a symmetry property of these velocities, the lengthy algebra usually involved in these studies can be eliminated, and it becomes feasible to examine the higher order terms in the expansion for the mean emf which supports the axially symmetric field. Braginskiĭ proved that, for his ordering of the velocity, this class of flows cannot give dynamo action at large magnetic Reynolds number. It is shown here that by increasing the magnitude of the asymmetric flow relative to that of the differential rotation, this ‘anti-dynamo’ restriction may be removed. The implications for numerical dynamo models are discussed.