Images of the giant planets Jupiter and Saturn show highly turbulent storms and swirling clouds that reflect the intensity of turbulence in their atmospheres. Quantifying planetary turbulence is inaccessible to conventional tools, however, since they require large quantities of spatially and temporally resolved data. Here we show, using experiments, observations, and simulations, that potential vorticity (PV) is a straightforward and universal diagnostic that can be used to estimate turbulent energy transfer in a stably stratified atmosphere. We use the conservation of PV to define a length scale, LM, representing a typical distance over which PV is mixed by planetary turbulence. LM increases as the turbulent intensity increases and can be estimated from any latitudinal PV profile. Using this principle, we estimate LM within Jupiter's and Saturn's tropospheres, showing for the first time that turbulent energy transfer in Saturn's atmosphere is four times less intense than Jupiter's.