Extreme vorticity events in turbulent Rayleigh-B\'enard convection from stereoscopic measurements and reservoir computing

High-amplitude events of the out-of-plane vorticity component $\omega_z$ are analyzed by stereoscopic particle image velocimetry (PIV) in the bulk region of turbulent Rayleigh-B\'{e}nard convection in air. The Rayleigh numbers ${\rm Ra}$ vary from $1.7 \times 10^4$ to $5.1 \times 10^5$. The experimental investigation is connected with a comprehensive statistical analysis of long-term time series of $\omega_z$ and individual velocity derivatives $\partial u_i/\partial x_j$. A statistical convergence for derivative moments up to an order of 6 is demonstrated. Our results are found to agree well with existing high-resolution direct numerical simulation data in the same range of parameters, including the extreme vorticity events which appear in the far exponential tails of the corresponding probability density functions. The transition from a Gaussian to a non-Gaussian velocity derivative statistics in the bulk of a convection flow is confirmed experimentally. The experimental data are used to train a reservoir computing model, one implementation of a recurrent neural network, to reproduce highly intermittent experimental time series of the vorticity and thus reconstruct extreme out-of-plane vorticity events. After training the model with high-resolution PIV data, the machine learning model is run with sparsely seeded, continually available, and unseen measurement data in the reconstruction phase. The dependence of the reconstruction quality on the sparsity of the partial observations is also documented. Our latter result paves the way to machine--learning--assisted experimental analyses of small-scale turbulence for which time series of missing velocity derivatives can be provided by generative algorithms.