Kelvin’s theorem applied to the Earth’s core

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We examine the consequences of adopting simplifying assumptions of a highly elec­trically conducting core and an insulating mantle for the magnetic field at the top of the core. These approximations form a zeroth-order model for the Earth, but may be quite accurate. We find that in addition to the well-known invariants which apply to patches of radial magnetic flux on the core surface which are bounded by con­tours on which the radial field vanishes, additional constraints apply to the same patches. These constraints are precisely those related to Kelvin’s theorem, which is well-known in rotating systems when magnetic fields are absent. In principle, the constraints can be tested by repeated observations of the field through time.