Entropy budget and coherent structures associated with a spectral closure model of turbulence

Abstract

We ‘derive’ the eddy-damped quasi-normal Markovian model (EDQNM) by a method that replaces the exact equation for the Fourier phases with a solvable stochastic model, and we analyse the entropy budget of the EDQNM. We show that a quantity that appears in the probability distribution of the phases may be interpreted as the rate at which entropy is transferred from the Fourier phases to the Fourier amplitudes. In this interpretation, the decrease in phase entropy is associated with the formation of structures in the flow, and the increase of amplitude entropy is associated with the spreading of the energy spectrum in wavenumber space. We use Monte Carlo methods to sample the probability distribution of the phases predicted by our theory. This distribution contains a single adjustable parameter that corresponds to the triad correlation time in the EDQNM. Flow structures form as the triad correlation time becomes very large, but the structures take the form of vorticity quadrupoles that do not resemble the monopoles and dipoles that are actually observed.

Two-Dimensional Flow on the Sphere

“…The sign change between (31) and (32) reflects the change in handedness. To see that this makes sense, recall that the first equation in (31) and ( 32) also applies to a passive scalar advected by the velocity field represented by ψ.…”

“…A second primary purpose is to illustrate the advantages of stereographic coordinates for both analytic and numerical work on the sphere. In stereographic coordinates, the equations governing two-dimensional turbulence on the sphere take a form, (31) or ( 37)-(39), that is very similar to the corresponding formulation in Cartesian coordinates on the plane. Only the appearance of the smoothly varying metric coefficient h distinguishes the two dynamics.…”

Entropy and Turbulence Structure

“…For determination of the turbulence energy spectra, for example, the well-known Kolmogorov theory [ 13 ] starts with the energy content at a given scale, then uses dimensional arguments to construct a power-law type of energy transfer in the so-called ‘inertial range”. Inter-scale transport theories, such as the eddy-damped quasi-normal Markovian (EDQNM) theory, introduce eddy-to-eddy momentum transport models to complete the spectral closure [ 14 , 15 ]. There exists a set of works in the literature invoking the maximum-entropy principle to derive the turbulence energy distribution; however, they again rely on the scale-to-scale transport concept to constrain the spectral function.…”

Scaling of the maximum-entropy turbulence energy spectra

“…Much effort has been expended on identifying the interaction mechanisms between the eddies at different scales. Some analytical methods have been developed since long ago, as in energy scaling in the inertial range ("k -5/3 -law" [1]), in two-dimensional turbulence [2], and a rather sophisticated method called EDQNM (eddy-damped quasi-normal Markovian) [3]. The references are intended as examples, among numerous others, and more complete reviews are available in the literature, e.g.…”

“…This functional form is intuitive and useful as it uses the basic turbulence variables to parameterize the energy distribution. Direct-interaction approximation [2] and EDQNM [3] tend to be complex in derivation and in the final form, reducing its accessibility. A compact, easily understandable theory is preferred, from aesthetic and application perspectives.…”