Probability density of displacement and overturning length scales under diverse stratification
 Vertical overturns, produced by turbulence in density-stratified lakes and oceans, are often quantified by the Thorpe scale, L-T. The correlation between L-T and the Ozmidov (energy-containing) scale can be used to estimate rates of turbulent dissipation and vertical diffusivities. Based on temperature microstructure measurements from several stratified lakes, the probability density functions (pdf's) of Thorpe displacements and overturning length scales are distinguished and discussed. The stratification was varying over five orders of magnitude between the different data sets, resulting in Thorpe scales between 1 cm and 100 m. It is shown that the analyzed pdf's follow a universal form that can be empirically described by an exponential function and parameterized by a single length scale. The functional form of the pdf of overturning length scales can be related to the inertial subrange, which is found in temperature, as well as in length-scale fluctuation spectra. As a result, the ratio between the maximum displacement length scale, L-max, and the Thorpe scale depends on the rate of turbulent kinetic energy dissipation. Based on the universal pdf of overturning length scales, we show with numerical simulations that L-max is a more appropriate macroscopic length scale than L-T for the estimation of turbulence based on displacements in temperature profiles.