Direct numerical simulation is used to investigate the integral behaviour of buoyant plumes subjected to a uniform crosswind. We find that the plume centreline is not aligned with the streamlines as is commonly assumed in integral plume models. Instead, the centreline is aligned with the total buoyancy flux, which implies that a model for the vertical turbulent buoyancy flux is required to faithfully predict the plume angle. A study of the volume conservation equation shows that the Leibniz term, i.e. entrainment due to incorporation of ambient fluid with non-zero velocity, is the dominant entrainment mechanism in strong crosswinds. The data indicate that pressure differences between the top and bottom of the plume play a leading role in the evolution of the horizontal and vertical momentum balances and are crucial to appropriately model plume rise. By parameterising the vertical buoyancy flux, the entrainment and the pressure, a novel integral plume model is developed which is in good agreement with the simulations for sufficiently strong crossflow.