A dynamically driven mechanism occurring above severe thunderstorms is described, which can explain the jumping cirrus phenomenon. A three-dimensional, non-hydrostatic cloud model is used to perform numerical simulation of a supercell that occurred in Montana in 1981. The jumping cirrus phenomenon is reproduced in the simulated storm. Analysis of the model results shows that the jumping cirrus phenomenon is produced by the breaking of the gravity waves excited by the strong convection inside the storm. The wave breaking process causes some moisture to detach from the storm cloud and jump into the stratosphere. The apparent upstream motion of the jumping cirrus is true only relative to the storm. The jumping cirrus phenomenon represents an irreversible transport mechanism of materials from the troposphere to the stratosphere.