Among the theorems of applied mathematics Cowling's theorem on the impossibility of axisymmetric homogeneous dynamos has played a special role because it has cast in doubt for 40 years the dynamo hypothesis of the origin of geomagnetism. This hypothesis goes back to Larmor (1919; see Larmor, J.) who argued that the magnetic field of sunspots could be created by motions in an electrically conducting fluid. He envisioned an axisymmetric model where radial motion into a cylindrical tube of magnetic flux would cause an electromotive force 
driving an electric current in the azimuthal direction, which in turn would enhance the magnetic flux in the cylinder. While this model appears to be superficially convincing, Cowling was able to prove that such a dynamo process is not possible.
The proof originally provided by Cowling (1934; see Cowling, T.G.) is quite simple since he was only concerned with the possibility of the steady generation of a field confined to meridional planes as envisioned...
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Busse, F. (2007). Cowling's Theorem. In: Gubbins, D., Herrero-Bervera, E. (eds) Encyclopedia of Geomagnetism and Paleomagnetism. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-4423-6_56
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