Towards the Suitability of Information Entropy as an LES Quality Indicator

The Shannon entropy is a rigorous measure to evaluate the complexity in dynamical systems. Shannon entropy can be directly calculated from any set of experimental or numerical data and yields the uncertainty of a given dataset. Originating from information theory, the concept can be generalized from assessing the uncertainty in a message to any dynamical system. Following the concept of ergodicity, turbulence forms another class of dynamical systems, which is generally assessed using statistical measures. The quantification of resolution quality is a crucial aspect in assessing turbulent-flow simulations. While a vast variety of statistical measures for the evaluation of resolution is available, measures closer representing the dynamics of a turbulent systems, such as the Wasserstein metric or the Ljapunov exponent become popular. This study investigates how the Shannon entropy can lead to useful insights in the quality of turbulent-flow simulations. The Shannon entropy is calculated based on distributions, which enables the direct evaluation from unsteady flow simulations or by post-processing. A turbulent channel flow and a planar turbulent jet are used as validation tests. The Shannon entropy is calculated for turbulent velocity- and scalar-fields and correlations with physical quantities, such as turbulent kinetic energy and passive scalars, are investigated. It is shown that the spatial structure of the Shannon entropy can be related to flow phenomena. This is illustrated by the investigation of the entropy of the velocity fluctuations, passive scalars and turbulent kinetic energy. Grid studies reveal the Shannon entropy as a converging measure. It is demonstrated, that classical turbulent-kinetic-energy-based quality measures struggle with the identification of insufficient resolution, while the Shannon entropy has demonstrated potential to form a solid basis for LES quality assessment.

On Yaglom’s Law for the Interplanetary Proton Density and Temperature Fluctuations in Solar Wind Turbulence

In the past decades, there has been an increasing literature on the presence of an inertial energy cascade in interplanetary space plasma, being interpreted as the signature of Magnetohydrodynamic turbulence (MHD) for both fields and passive scalars. Here, we investigate the passive scalar nature of the solar wind proton density and temperature by looking for scaling features in the mixed-scalar third-order structure functions using measurements on-board the Ulysses spacecraft during two different periods, i.e., an equatorial slow solar wind and a high-latitude fast solar wind, respectively. We find a linear scaling of the mixed third-order structure function as predicted by Yaglom’s law for passive scalars in the case of slow solar wind, while the results for fast solar wind suggest that the mixed fourth-order structure function displays a linear scaling. A simple empirical explanation of the observed difference is proposed and discussed.

Impact of Along-Valley Orographic Variations on the Dispersion of Passive Tracers in a Stable Atmosphere

A numerical model is used to investigate the transport of passive tracers in an idealized Alpine valley during stable wintertime conditions after the evening transition. The valley is composed of an upstream-valley section, which opens on a narrower downstream valley section, which opens onto a plain. The ratio between the valley-floor widths of the upstream and downstream sections is either 4 (simulation P1) or 11.5 (P2). The change in the thermal structure of the atmosphere in the along-valley direction and over the plain leads to the development of an along-valley flow. This flow is up-valley in the upstream section during the first three hours of the P1 simulation, reversing to the down-valley direction afterwards, but remains up-valley during the six hours of the P2 simulation. The effect of wind dynamics on the dispersion of passive scalars is identified by tracking areas prone to stagnation, recirculation, and ventilation using the methodology developed by Allwine and Whiteman (1994). Zones identified as prone to stagnation are consistent with those of high tracer concentration in both simulations. The narrowing of the valley is found to significantly reduce ventilation in the upstream section, an observation quantified by a ventilation efficiency.