The interest in the organized part of turbulent flows has increased considerably over the last thirty years. This is particularly true in atmospheric convective boundary layers, where observed fields of coherent structures (CSs) are seen to evolve from a background of microscale turbulent fluctuations. These CSs are viewed as fundamental building blocks that grow and interact to help establish more semi-permanent large-scale structures in atmospheric flows. CSs are fundamentally important to gaining an understanding of turbulence in general. In spite of observational successes, however, the problem of describing turbulent systems with coherent structures remains as a formidable theoretical challenge.
In this research, low-order models (LOMs) for the dynamics of CSs in convective boundary layers will be developed. LOMs are commonly obtained by the Galerkin method and reveal basic mechanisms and their interplay through the focus on key elements, retaining only minimal number of degrees of freedom. In this method, fluid dynamical fields are expanded into infinite sets of time-independent basis functions; then projection of the original partial differential equations onto these functions yields a finite system of ordinary differential equations (the LOM) for the time evolution of the coefficients in truncated expansions. Following Lumley and coworkers, empirical orthogonal functions (EOFs) will be used as the basis functions. These will be computed from large eddy simulations (LES) and (potentially) observational data.
This research will also solve two important problems that will allow more physically sound LOMs. These are: 1) The Galerkin method provides no guarantee against violations (in the LOM) of fundamental conservation properties of the original equations, which often results in unphysical behavior. 2) The EOFs are characteristics of the underlying probability distribution, estimated from data and, thus, subject to sampling errors. These should be appropriately quantified before interpreting and using sample EOFs. This will be accomplished by employing improvements over the traditional Galerkin approximations developed by the Co-PIs in their previous research.
Broader impacts of the proposed activity The (re)discovery of CSs has led to questioning the relevance of the traditional statistical theory of turbulence. The problem of describing turbulent systems with coherent structures presents a formidable theoretical challenge. This research, resulting in relatively simple gyrostatic LOMs inherently possessing fundamental conservation properties of the fluid dynamic equations, will provide a new effective tool for scientific understanding of essential mechanisms and their interaction in CSs dynamics. This research should broadly impact the understanding of turbulent phenomena in general, as well as the intricacies of convective organization in atmospheric boundary layer flows. Further, this work impacts the general understanding of nature of the interacting roles of statistics and nonlinear dynamics.
