On the robustness of the toroidal velocity theorem

摘要

In dynamo theory the toroidal velocity theorem in its classical version (Elsasser, Phys. Rev. 1946, vol. 69, pp. 106-116, Bullard and Gellman, Phil. Trans. R. Soc. Lond. 1954, vol. A247, pp. 213-278) rules out dynamo action in a spherical conducting volume provided that the fluid is incompressible, the conductivity is uniform, and the velocity field is purely toroidal. We prove in this note that this result is robust in the sense that slight compressibility of the fluid, small non-radial variations and even large radial variations in conductivity, and the presence of a small non-toroidal velocity component do not invalidate the theorem. Moreover, by proper choice of the conductivity distribution modelling the conducting volume, small deviations from spherical symmetry of the conductor can also be taken into account.