A method of visualizing and characterizing stirring structures of high Rayleigh number geophysical flows whose time dependence is strongly aperiodic is presented. To this end, the system is leaked by defining a smaller region of the flow, so that a particle is considered to be escaped if it enters this region. By means of an ensemble of nonescaped tracers, we are able to characterize stirring and transport processes by visualizing the converging and stretching filamentations (stable and unstable manifolds) in the flow. The method indicates that the present-day Earth's mantle is not well stirred because the time that has passed since the formation of the Earth has not been long enough for the flow of the mantle to generate efficient stirring, and observations reveal indeed the existence of reservoirs of different materials.