On a solution of the closure problem for dry convective boundary layer turbulence and beyond

We consider the closure problem of representing the higher order moments (HOMs) in terms of lower-order moments, a central feature in turbulence modelling based on the Reynolds-Averaged Navier-Stokes (RANS) approach. Our focus is on models suited for the description of asymmetric, non-local and semi-organized turbulence in the dry atmospheric convective boundary layer (CBL). We establish a multivariate probability density function (PDF) describing populations of plumes which are embedded in a sea of weaker randomly spaced eddies, and apply an assumed Delta-PDF approximation. The main content of this approach consists of capturing the bulk properties of the PDF. We solve the closure problem analytically for all relevant higher order moments (HOMs) involving velocity components and temperature and establish a hierarchy of new non-Gaussian turbulence closure models of different content and complexity ranging from analytical to semi-analytical. All HOMs in the hierarchy have a universal and simple functional form. They refine the widely used Millionshchikov closure hypothesis and generalize the famous quadratic skewness-kurtosis relationship to higher-order. We examine the performance of the new closures by comparison with measurement, LES and DNS data and derive empirical constants for semi-analytical models, which are best for practical applications. We show that the new models have a good skill in predicting the HOMs for atmospheric CBL. Our closures can be implemented in second-, third- and fourth-order RANS turbulence closure models of bi-, tri-and four-variate levels of complexity. Finally, several possible generalizations of our approach are discussed.

Quasi-Entropy Closure: A Fast and Reliable Approach to Close the Moment Equations of the Chemical Master Equation

MotivationThe Chemical Master Equation is the most comprehensive stochastic approach to describe the evolution of a (bio-)chemical reaction system. Its solution is a time-dependent probability distribution on all possible configurations of the system. As the number of possible configurations is typically very large, the Master Equation is often practically unsolvable. The Method of Moments reduces the system to the evolution of a few moments of this distribution, which are described by a system of ordinary differential equations. Those equations are not closed, since lower order moments generally depend on higher order moments. Various closure schemes have been suggested to solve this problem, with different advantages and limitations. Two major problems with these approaches are first that they are open loop systems, which can diverge from the true solution, and second, some of them are computationally expensive.ResultsHere we introduce Quasi-Entropy Closure, a moment closure scheme for the Method of Moments which estimates higher order moments by reconstructing the distribution that minimizes the distance to a uniform distribution subject to lower order moment constraints. Quasi-Entropy closure is similar to Zero-Information closure, which maximizes the information entropy. Results show that both approaches outperform truncation schemes. Moreover, Quasi-Entropy Closure is computationally much faster than Zero-Information Closure. Finally, our scheme includes a plausibility check for the existence of a distribution satisfying a given set of moments on the feasible set of configurations. Results are evaluated on different benchmark problems.Abstract Figure

A simple model for interpreting temperature variability and its higher-order changes

Atmospheric temperature distributions are often identified with their variance, while the higher-order moments receive less attention. This can be especially misleading for extremes, which are associated with the tails of the Probability Density Functions (PDFs), and thus depend strongly on the higher-order moments. For example, skewness is related to the asymmetry between positive and negative anomalies, while kurtosis is indicative of the ”extremity” of the tails. Here we show that for near-surface atmospheric temperature, an approximate linear relationship exists between kurtosis and skewness squared. We present a simple model describing this relationship, where the total PDF is written as the sum of three Gaussians, representing small deviations from the climatological mean together with the larger amplitude cold and warm temperature anomalies associated with synoptic systems. This model recovers the PDF structure in different regions of the world, as well as its projected response to climate change, giving a simple physical interpretation of the higher-order temperature variability changes. The kurtosis changes are found to be largely predicted by the skewness changes. Building a deeper understanding of what controls the higher-order moments of the temperature variability is crucial for understanding extreme temperature events and how they respond to climate change.

Higher Order Moments of the Order Statistics for the Rectangular, Exponential, Gamma and Weibull Distributions

In this paper we discuss the problem of Higher Order Moments for the order Statistics for the Rectangular, Exponential, Gamma and Weibull distributions by finding the order statistic distributions for the base distribution and modified distributions, the base distribution is to deduce the corresponding distribution by the polynomial modifier. These higher order moments are very much useful in most of the Data sciences and Image analysis.