Modern observations of the geomagnetic field reveal fluctuations(1-3) with a dominant period of about 60 years. These fluctuations are probably a result of waves in the liquid core(4), although the precise nature of the waves is uncertain. Common suggestions include a type of magnetic wave, known as a torsional oscillation(5), but recent studies(6) favour periods that are too short to account for a 60-year fluctuation. Another possibility involves MAC waves(7), which arise from the interplay between magnetic, Archimedes and Coriolis forces. Waves with a suitable period can emerge when the top of the core is stably stratified. Here I show that MAC waves provide a good description of time-dependent zonal flow at the top of the core(8), as inferred from geomagnetic secular variation(9). The same wave motion can also account for unexplained fluctuations in the dipole field(10). Both of these independent predictions require a 140-kilometre-thick stratified layer with a buoyancy frequency comparable to the Earth's rotation rate. Such a stratified layer could have a thermal origin(11,12), implying a core heat flow of about 13 terawatts. Alternatively, the layer could result from chemical stratification(13,14). In either case, the existence of a stratified layer at the top of the core obscures the nature of flow deeper in the core, where the magnetic field is continually regenerated.