Simple core undertones

Summary. If undertones (internal gravity waves in the Earth's liquid outer core) exist then they occur at such long periods that the full theory of Earth dynamics in a rotating reference frame is required for their description. This theory has been so cumbersome to implement that very severe truncation of the traditional vector spherical harmonic expansions has been required, with the consequence that the results obtained have been noted less for their physical significance than for their technical achievement. By considering simplified models of the Earth's core it is first demonstrated that, at least for the case of a non‐rotating Earth, the effects of mantle structure and boundary rigidity on the undertone periods are minimal. Further, the combined effects of core compressibility and self‐gravitation in a real Earth model can to a good approximation be replaced by a single parameter, the Brunt–Väisälä buoyancy frequency, in a simple incompressible fluid model. On applying these simplifications to a rotating Earth model, the resulting equations of motion are then solved with a longer coupling chain than any used previously. This approach, based on the chain beginning with the spheroidal term S 2 2 , supports the hypothesis that the periods of the core undertones for a realistic rotating Earth model are crowded into a narrow band around 12 hr.