On the applicability of the frozen flux approximation in core flow modelling as a function of temporal frequency and spatial degree

The secular variation of the geomagnetic field is due to flow of molten iron in the Earth's core. One can invert for the flow at the core-surface, assuming that the magnetic field is frozen into the core fluid. The validity of the frozen flux approximation (FFA) depends on the scale length of the magnetic field and the period of core motions. A first-order quantitative assessment can be made by solving the induction equation for a spherical conductor oscillating in an ambient magnetic field. The period, at which the FFA holds, strongly decreases with increasing spherical harmonic degree of the ambient field. For degrees smaller than five, the flux of the ambient magnetic field can be considered as frozen-in for periods of more than a thousand years. For ambient magnetic fields of spherical harmonic degree 10 and higher, the FFA still holds up to periods of about a hundred years. However, the real situation in the core could be more complex if the field cannot be assumed as ambient but is generated by the flow itself.