Probabilistic data-driven turbulence closure modeling by assimilating statistics

A framework for deriving probabilistic data-driven closure models is proposed for coarse-grained numerical simulations of turbulence in statistically stationary state. The approach unites the ideal large-eddy simulation model and data assimilation methods. The method requires a posteriori measured data to define stochastic flow perturbations, which are combined with a Bayesian statistical correction enforcing user-specified statistics extracted from high-fidelity flow snapshots. Thus, it enables computationally cheap ensemble simulations by combining knowledge of the local integration error and knowledge of desired flow statistics. A model example is given for two-dimensional Rayleigh-B\'enard convection at Rayleigh number $\mathit{Ra}=10^{10}$, incorporating stochastic perturbations and an ensemble Kalman filtering step in a non-intrusive way. Physical flow dynamics are obtained, whilst kinetic energy spectra and heat flux are accurately reproduced in long-time ensemble forecasts on coarse grids. The model is shown to produce accurate results with as few as 20 high-fidelity flow snapshots as input data.