An Optimal Parametrization of Turbulent Scales

Abstract:

To numerically capture the large-scale dynamics of atmospheric flows, geophysicists need to rely on reasonable parametrizations of the energy transfers to and from the non-resolved small scale eddies, mediated through turbulence. The task is notoriously not trivial, and is typically solved by ingenious but ad-hoc elaborations on the concept of eddy viscosities.
The difficulty is tied into the intrinsic Non-Gaussianity of turbulence, a feature that may explain why standard Quasi-Normal cumulant discard statistical closure strategies can fail dramatically, an example being the development of negative energy spectra in Millionshtchikov's 1941 Quasi-Normal (QN) theory. While Orszag's 1977 Eddy Damped Quasi Normal Markovian closure (EDQNM) provides an ingenious patch to the issue, the reason why the QN theory fails so badly is not so clear. Are closures necessarily either trivial or ad-hoc, when proxies for true ensemble averages are taken to be Gaussian ? The purpose of the talk is to answer negatively, using the lights of a new ``optimal closure framework'' recently exposed by [Turkington,2013]. For turbulence problems, the optimal closure allows a consistent use of a Gaussian Ansatz (and corresponding vanishing third cumulant) that also retains an intrinsic damping. The key to this apparent paradox lies in a clear distinction between the true ensemble averages and their proxies, most easily grasped provided one uses the Liouville equation as a starting point, rather than the cumulant hierarchy. Schematically said, closure is achieved by minimizing a lack-of-fit residual, which retains the intrinsic features of the true dynamics.
The optimal closure is not restricted to the Gaussian modeling. Yet, for the sake of clarity, I will discuss the optimal closure on a problem where it can be entirely implemented, and compared to DNS : the relaxation of an arbitrarily far from equilibrium energy shell towards the Gibbs equilibrium for truncated Euler dynamics. Predictive insights are obtained, among which a non-trivial universal relaxation profile for the inverse temperatures (see associated figure). It is associated to a non-standard dissipation operator responsible for the turbulent damping, possibly relevant for atmospheric dynamics.