A large part of the Earth's magnetic field is generated by fluid motion in the molten outer core. As a result of continuous satellite measurements since 1999, the core magnetic field and its recent variations can now be described with a high resolution in space and time. These data have recently been used to investigate small-scale core flow, but no advantage has yet been taken of the improved temporal resolution, partly because the filtering effect of the electrically conducting mantle was assumed to mask short-period magnetic variations. Here we show that changes in the magnetic field occurring over only a few months, indicative of fluid flow at the top of the core, can in fact be resolved. Using nine years of magnetic field data obtained by satellites as well as Earth-based observatories, we determine the temporal changes in the core magnetic field and flow in the core. We find that the core flow is spatially localized and involves rapid variations over a few months, with surprisingly large local accelerations. Our results suggest that short-term fluctuations of the core magnetic field are robust features of rapid core dynamics and should be considered in the development of future numerical models of the geodynamo.