Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous

TL;DR

This paper reviews advances in statistical turbulence modeling, focusing on inhomogeneous closures and subgrid models for geophysical flows, highlighting developments that improve realism and computational efficiency.

Contribution

It introduces new inhomogeneous closure models like EDMAC and EDMIC, extending turbulence theory to complex real-world interactions and improving simulation accuracy.

Findings

Development of realizable inhomogeneous closures like QDIA and MIC.

Formulation and testing of efficient EDMAC and EDMIC models.

Application of subgrid models to geophysical flows with Rossby waves and topography.

Abstract

Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was again based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian inhomogeneous closures (MICs). Recent developments include the formulation and testing of an eddy damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy damped quasi-normal Markovian (EDQNM). As well a similarly efficient closure, the realizable eddy damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields.