Krachnan derived the equations for his Eulerian direct interaction approximation (DIA) closure for homogeneous isotropic turbulence based on formal renormalized perturbation theory in 1959. His work had elements in common with the renormalized perturbation theory and functional approaches to quantum electrodynamics (QED), famously developed by Feynman, Schwinger and Tomonaga, for which they were awarded the 1965 Nobel Prize in physics. Unlike QED, where the fine structure constant, measuring interaction strength is only ∼1/137, turbulence at high Reynolds number is a problem of strong interaction. Formally, the DIA is a second order (two-point) non-Markovian (time-history) closure for the renormalized two-time covariances or cumulants and response functions where the interaction coefficients, or vertices, are unrenormalized or bare.
The subsequent works of Frederiksen and O’Kane led to the development and computational implementation of the quasi-diagonal direct interaction approximation (QDIA) closure for inhomogeneous turbulence interacting with mean flows, waves and topography. The QDIA is again a non-Markovian closure which has the important property of being tractable but nevertheless represents a significant computational challenge in that, like the DIA, information about the past evolution is integrated over. The QDIA closure has proved remarkably accurate in application to a wide range of problems in dynamics, predictability, data assimilation and subgrid modelling for inhomogeneous flows and has been formulated for general classical and quantum field theories.
In a recent research work, published in Journal of Fluid Mechanics, CSIRO Oceans and Atmosphere scientists in Australia, Professors Jorgen S. Frederiksen and Terence J. O’Kane, conducted a study aimed at making the QDIA closure more computationally efficient. The researchers’ goal was to develop a Markovian version of the QDIA closure for inhomogeneous turbulence with similar efficiency to the eddy-damped quasi-normal Markovian closure (EDQNM) of Orszag and the realizable Markovian closure (RMC) of Bowman for homogeneous turbulence. They developed three versions of their Markovian inhomogeneous closure (MIC) for the problem of general two-dimensional inhomogeneous turbulent flows interacting with Rossby waves and topography.
Briefly, the research team prepared a detailed summary of the equations for general two-dimensional flows over topography on a generalized β-plane with doubly periodic boundary conditions. Next, the QDIA closure model was presented closely followed by the derivation and formulation of three versions of the MIC based on a Markovian form of the response function and employing three commonly used versions of the fluctuation-dissipation theorem (FDT) for the two-time covariance. Lastly, they compared the performance of the three versions of the MIC with two variants of the non-Markovian QDIA (with and without cumulant update restarts) and an ensemble of 1800 direct numerical simulations (DNS).
The researchers found that the derived Markov equations for the triad relaxation functions carried similar information to the time-history integrals of the non-Markovian QDIA closure and noted the improved efficiency for long time integrations, T, i.e. O(T) versus O(T3). They studied the ‘far from equilibrium process’ in which a westerly mean flow impacted on a conical mountain and generated large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Remarkably, in all the cases, the pattern correlations between the evolved mean Rossby wavetrains in the closures and the DNS ensemble were greater than 0.9998.
In summary, the Frederiksen-O’Kane study successfully presented the formulation of manifestly MIC models for turbulent flows and Rossby waves over topography on a generalized β-plane. The three different MICs used employed the current-time FDT (quasi-stationary), the prior-time FDT (non-stationary) and the correlation FDT, respectively. To sum up, excellent agreement between the evolved mean stream function and mean and transient kinetic energy spectra were found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. This work constitutes a basis for deriving analytic expressions for the unique MIC triad relaxation rates and further gains in computational efficiency and higher resolution studies of turbulent geophysical flows.
Jorgen S. Frederiksen, Terence J. O’Kane. Markovian inhomogeneous closures for Rossby waves and turbulence over topography. Journal of Fluid Mechanics (2019), volume 858, pages 45–70.Go To Journal of Fluid Mechanics