Counting the structures in turbulent flows

The baselines against which turbulent flow statistics are compared often take the form of simple statistical distributions, such as Gaussian or log-normal. The need for such strong and simple baselines has been recognized since the earliest theoretical turbulence works, if for no other reason than to clarify what is meant by claiming that some statistical properties of turbulence are "anomalous." As turbulence analysis has matured from two-point correlation statistics and has, facilitated by direct numerical simulation, advanced into the quantitative analysis of coherent structures, there is a apparent need for new null hypotheses. In this work, we study the excursion sets of Gaussian random fields as baseline cases for comparing to turbulent coherent structures. Using the well-developed theory for the topology of excursion sets, we derive expressions for the number of energy-containing structures in a triply-periodic turbulent flow. We find that at large scales, homogeneous and isotropic turbulence behaves as a Gaussian random field, as expected. At small scales, while there is significant deviation for a range of excursion set thresholds, we find that intense energy-containing structures follow the predictions of Gaussian random field theory surprinsingly well, allowing us to accurately predict the number of such structures well beyond the precision of percolation theory.