Describing the statistics of turbulent flow by using the principle of maximum entropy

Burgers and Onsager were pioneers in using statistical mechanics in the theory of turbulent fluid motion. Their approach was, however, rather different. Whereas Onsager stayed close to the energy conserving Hamiltonian systems of classical mechanics, Burgers explicitly exploited the fact that turbulent motion is forced and dissipative. The basic assumption of Burgers' approach is that forcing and dissipation balance on average, an assumption that leads to interesting conclusions concerning the statistics of turbulent flow but also to a few problems. A compilation and assessment of his work can be found in [1].

We have taken up the thread of Burgers' approach and rephrased it in terms of Jaynes' principle of maximum entropy. The principle of maximum entropy yields a  statistical description in terms of a probability density function that is as noncommittal as possible while reproducing any given expectation values. In the spirit of Burgers, these expectation values are the average energy as well as the average of the first and higher order time-derivatives of the energy (or other global quantities). In [2] the method was applied to a system devised by Lorenz . By using constraints on the average energy and its first and second order time-derivatives a satisfying description was produced of the system's statistics, including covariances between the different variables. 

Burgers' approach can also be applied to the parametrization problem, i.e., the problem of how to deal statistically with scales of motion that cannot be resolved explicitly. Quite recently we showed this for two-dimensional turbulence on a doubly periodic flow domain, a system that is relevant as a first-order approximation of large-scale balanced flow in the atmosphere and oceans. Using a spectral description of the system it is straightforward to separate between resolved and unresolved scales and by using a reference model with high resolution it is possible to study how well a parametrization performs by implementing it in the same model with a lower resolution. Based on two studies [3, 4] we will show how well the principle of maximum entropy works in tackling the problem of unresolved turbulent scales.  

[1] F.T.M. Nieuwstadt and J.A. Steketee, Eds., 1995: Selected Papers of J.M. Burgers. Kluwer Academic, 650 pp. 

[2] W.T.M. Verkley and C.A. Severijns, 2014: The maximum entropy principle applied to a dynamical system proposed by Lorenz. Eur. Phys. J. B, 87:7, https://doi.org/10.1140/epjb/e2013-40681-2 (open access).  

[3] W.T.M. Verkley, P.C. Kalverla and C.A. Severijns, 2016: A maximum entropy approach to the parametrization of subgrid processes in two-dimensional flow. Quarterly Journal of the Royal Meteorological Society, 142, 2273-2283, https://doi.org/10.1002/qj.2817

[4] W.T.M. Verkley, C.A. Severijns and B.A. Zwaal, 2019: A maximum entropy approach to the interaction between small and large scales in two-dimensional turbulence. Quarterly Journal of the Royal Meteorological Society, 145, 2221-2236, https://doi.org/10.1002/qj.3554