The strongest supercell thunderstorms typically feature an above-anvil cirrus plume (AACP), which is a plume of ice and water vapor in the lower stratosphere that occurs downwind of the ambient stratospheric flow in the lee of overshooting deep convection. AACP-origin hydration of the stratosphere has a poorly constrained role in ozone destruction and surface warming. In this study, we use large eddy simulations corroborated by radar observations to understand the physics of AACP generation. We show that the overshooting top of a simulated supercell can act as a topographic obstacle and drive a hydraulic jump downstream at the tropopause, similar to a windstorm moving down the slope of a mountain but without solid topography. Once the jump is established, water vapor injection deep into the stratosphere may exceed 7 tonnes per second.