Direct numerical simulations of potential enstrophy in strongly stratified, weakly rotating turbulence will be presented. When rotation and stratification are strong - as in large-scale atmospheric and oceanic flows - the potential vorticity is approximately linear in the flow variables, and so the potential enstrophy is a quadratic invariant. At smaller scales, however, the quadratic approximation becomes questionable. We will show that both quadratic and higher-order potential enstrophy are possible in this regime, depending on the effects of viscosity. As a result, laboratory and geophysical scale stratified turbulence have very different potential enstrophy dynamics.