An extension of the toroidal theorem

The well-known toroidal theorem of Elsasser and Bullard and Gellman rules out dynamo action in a conducting sphere when the velocity field has no poloidal part. It is here shown that for a fixed toroidal velocity field any poloidal velocity must attain a finite size if dynamo action is to be possible. The resulting anti-dynamo theorem generalises the earlier result of Childress by giving a bound on the product of the suprema of the toroidal and poloidal velocities.