Plasma structures, here typified by the term 'plasmoids', in the solar wind impacting on the magnetopause, i.e. the boundary between the solar wind and the Earth's magnetosphere, can penetrate this boundary and be injected into the magnetosphere. This can happen either by expulsion of the magnetic field from the structure and subsequent diffusion of the magnetic field into the structure or by the formation of a polarization electric field that lets the plasma structure E × B-drift into the earth's magnetic field. In both cases a collisionless resistivity is required at some stage of the process. While magnetic expulsion requires electromagnetic models for its description, polarization can be modelled electrostatically and both processes can be, and have been, studied in laboratory experiments.We present three-dimensional electrostatic particle-in-cell simulations that reproduce large-amplitude waves, in the lower-hybrid range, that have been observed in laboratory experiments. Lower-hybrid waves have also been seen at the magnetopause of the earth. We consider the implications for spacecraft-based studies of magnetopause penetration, and suggest that the search for penetrating plasma structures should emphasize cases in which the interplanetary magnetic field is oriented northwards, as this configuration is less likely for reconnection. The application of theoretical predictions to the magnetopause environment shows that a plasma structure penetrating via polarization needs to be small, i.e. less than 10–100 km wide for typical parameters, and that wave processes at the magnetopause are needed to create such small structures. A larger structure can penetrate by means of magnetic expulsion.