Synchronization to Big Data: Nudging the Navier-Stokes Equations for Data Assimilation of Turbulent Flows

Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it to the canonical problem of fluid dynamics: three-dimensional homogeneous and isotropic turbulence. By doing numerical experiments we perform a systematic assessment of how well the technique reconstructs large- and small-scale features of the flow with respect to the quantity and the quality or type of data supplied to it. The types of data used are (i) field values on a fixed number of spatial locations (Eulerian nudging), (ii) Fourier coefficients of the fields on a fixed range of wave numbers (Fourier nudging), or (iii) field values along a set of moving probes inside the flow (Lagrangian nudging). We present state-of-the-art quantitative measurements of the scale-by-scale transition to synchronization and a detailed discussion of the probability distribution function of the reconstruction error, by comparing the nudged field and the truth point by point. Furthermore, we show that for more complex flow configurations, like the case of anisotropic rotating turbulence, the presence of cyclonic and anticyclonic structures leads to unexpectedly better performances of the algorithm. We discuss potential further applications of nudging to a series of applied flow configurations, including the problem of field reconstruction in thermal Rayleigh-Bénard convection and in magnetohydrodynamics, and to the determination of optimal parametrization for small-scale turbulent modeling. Our study fixes the standard requirements for future applications of nudging to complex turbulent flows.